Commit 0d6d39e6f4942e897114d66e96723f641648d0a9 - gap-design

+140

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CHANGES.md less more

	0	Main changes from DESIGN 1.6 to DESIGN 1.7 (March 2019)
	1	-------------------------------------------------------
	2
	3	1. Lists of changes moved from README to new file CHANGES.md.
	4
	5	2. README now README.md.
	6
	7	3. gpl.txt now LICENSE.
	8
	9	4. Added test directory tst, file tst/testall.g, and a test file
	10	tst/testall.tst.
	11
	12	5. DESIGN package is now hosted on github.
	13
	14	6. The utility function DESIGN_IntervalForLeastRealZero was documented.
	15
	16	7. Documentation updated to reflect move to github and version change.
	17
	18	Main change from DESIGN 1.5 to DESIGN 1.6 (November 2011)
	19	---------------------------------------------------------
	20
	21	1. Revised installation instructions.
	22
	23	Main changes from DESIGN 1.4 to DESIGN 1.5
	24	------------------------------------------
	25
	26	1. Added internal function:
	27	- DESIGN_IntervalForLeastRealZero
	28
	29	2. Added user functions:
	30	- PointBlockIncidenceMatrix
	31	- ConcurrenceMatrix
	32	- InformationMatrix
	33	- BlockDesignEfficiency
	34
	35	3. Updated documentation, including documentation for
	36	the new user functions.
	37
	38	Main changes from DESIGN 1.3 to DESIGN 1.4
	39	------------------------------------------
	40
	41	1. Added function: AGPointFlatBlockDesign.
	42
	43	2. Improved and documented the main features of function
	44	SemiLatinSquareDuals, which can be used to classify (n x n)/k semi-Latin
	45	squares and SOMA(k,n)s with given properties.
	46
	47	3. Improved efficiency of function PGPointFlatBlockDesign.
	48
	49	4. Made function TDesignLambdas faster, in particular in the case of
	50	a point-transitive design, when that design's record includes a known
	51	point-transitive subgroup of the design's automorphism group.
	52
	53	5. Improved the (undocumented) function IsPossiblySBIBD so that
	54	TDesignBlockMultiplicityBound now makes full use of the Bruck-Ryser-Chowla
	55	Theorem when t=2 and `k<v=b`.
	56
	57	6. Improved the speed of functions TDesignBlockMultiplicityBound and
	58	ResolvableTDesignBlockMultiplicityBound by employing binary search to
	59	find the bounds.
	60
	61	Main changes from DESIGN 1.2 to DESIGN 1.3
	62	------------------------------------------
	63
	64	1. Added function: WittDesign.
	65
	66	2. Fixed a bug which caused a break loop to be entered in certain cases
	67	in function BlockDesigns when the least element of blockSizes was equal
	68	to tSubsetStructure.t.
	69
	70	3. Fixed a rarely occurring bug which caused no designs to be returned
	71	in function BlockDesigns in certain cases when the designs had just one
	72	block each.
	73
	74	4. Small improvements made to the documentation and the code.
	75
	76	Main changes from DESIGN 1.1 to DESIGN 1.2
	77	------------------------------------------
	78
	79	1. Added functions:
	80	- BlockIntersectionPolynomial
	81	- BlockIntersectionPolynomialCheck
	82	- TDesignLambdas
	83	- TDesignLambdaMin
	84	- TDesignIntersectionTriangle
	85	- SteinerSystemIntersectionTriangle
	86	- TDesignBlockMultiplicityBound
	87	- ResolvableTDesignBlockMultiplicityBound
	88	- BlockDesignPoints
	89	- NrBlockDesignPoints
	90	- BlockDesignBlocks
	91	- NrBlockDesignBlocks
	92	- PairwiseBalancedLambda
	93	- AffineResolvableMu
	94	- DeletedPointsBlockDesign
	95	- DeletedBlocksBlockDesign
	96	- AddedPointBlockDesign
	97	- AddedBlocksBlockDesign
	98	- DerivedBlockDesign
	99	- ResidualBlockDesign
	100
	101	2. The function BlockDesign can now (optionally) construct a block
	102	design from base blocks and a group of automorphisms, and the list
	103	of blocks given to BlockDesign need no longer be sorted.
	104
	105	3. Improved documentation of the function TDesignFromTBD, and changed
	106	the parameters (in a backward compatible way) to match the description
	107	of the "*-construction" in the McSorley-Soicher paper.
	108
	109	4. The (undocumented) resolveSimple option removed from functions
	110	BlockDesigns and PartitionsIntoBlockDesigns, and the option b added
	111	to these functions (to specify the number of blocks in a required block
	112	design or partition part).
	113
	114	5. Improvements made to the use of store (and time?) by the function
	115	BlockDesigns.
	116
	117	Main changes from DESIGN 1.0 to DESIGN 1.1
	118	------------------------------------------
	119
	120	1. Made compatible with GAP 4.4 and its package loading mechanism.
	121	DESIGN 1.1 works only with GAP 4.4/GRAPE 4.2/GAPDoc.
	122
	123	2. Upgraded XML I/O to DTRS protocol 2.0.
	124
	125	3. BlockDesignsToXMLFile has additional parameters to allow more user
	126	control over expansion, output, and design id's. The default now is for
	127	minimal expansion and writing out, which is a change from DESIGN 1.0.
	128
	129	4. Added functions:
	130	- BlockDesignIsomorphismClassRepresentatives,
	131	- DualBlockDesign,
	132	- ComplementBlocksBlockDesign.
	133
	134	5. Function names made read-only.
	135
	136	6. Some utility functionality moved to GRAPE 4.2.
	137
	138	7. Added files README and gpl.txt.
	139

+340

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LICENSE less more

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+23

-119

PackageInfo.g less more

4	4
5	5	SetPackageInfo( rec(
6	6
7		## This is case sensitive, use your preferred spelling.
8	7	PackageName := "DESIGN",
	8	Subtitle := "The Design Package for GAP",
	9	Version := "1.7",
	10	Date := "18/03/2019", # dd/mm/yyyy format
	11	License := "GPL-2.0-or-later",
9	12
10		## See '?Extending: Version Numbers' in GAP help for an explanation
11		## of valid version numbers.
12		Version := "1.6",
	13	SourceRepository := rec(
	14	Type := "git",
	15	URL := "https://github.com/gap-packages/design",
	16	),
	17	IssueTrackerURL := Concatenation( ~.SourceRepository.URL, "/issues" ),
	18	PackageWWWHome := "https://gap-packages.github.io/design",
	19	README_URL := Concatenation( ~.PackageWWWHome, "/README.md" ),
	20	PackageInfoURL := Concatenation( ~.PackageWWWHome, "/PackageInfo.g" ),
	21	ArchiveURL := Concatenation( ~.SourceRepository.URL,
	22	"/releases/download/v", ~.Version,
	23	"/design-", ~.Version ),
13	24
14		## Release date of the current version in dd/mm/yyyy format.
15		Date := "23/11/2011",
	25	ArchiveFormats := ".tar.gz",
16	26
17		## URL of the archive(s) of the current package release, but without
18		## the format extension(s), like '.zoo', which are given next.
19		## The archive file name must be changed with each version of the archive
20		## (and probably somehow contain the package name and version).
21		ArchiveURL := "http://www.maths.qmul.ac.uk/~leonard/design/design1r6/design1r6",
22
23		## All provided formats as list of file extensions, separated by white
24		## space or commas.
25		## Currently recognized formats are:
26		## .zoo the (GAP-traditional) zoo-format with "!TEXT!" comments
27		## for text files
28		## .tar.gz the UNIX standard
29		## .tar.bz2 compressed with 'bzip2', often smaller than with gzip
30		## -win.zip zip-format for DOS/Windows, text files must have DOS
31		## style line breaks (CRLF)
32		##
33		## In the future we may also provide .deb or .rpm formats which allow
34		## a convenient installation and upgrading on Linux systems.
35		##
36		ArchiveFormats := ".tar.gz", # the others are generated automatically
37
38		## If not all of the archive formats mentioned above are provided, these
39		## can be produced at the GAP side. Therefore it is necessary to know which
40		## files of the package distribution are text files which should be unpacked
41		## with operating system specific line breaks. There are the following
42		## possibilities to specify the text files:
43		##
44		## - specify below a component 'TextFiles' which is a list of names of the
45		## text files, relative to the package root directory (e.g., "lib/bla.g")
46		## - specify below a component 'BinaryFiles' as list of names, then all other
47		## files are taken as text files.
48		## - if no 'TextFiles' or 'BinaryFiles' are given and a .zoo archive is
49		## provided, then the files in that archive with a "!TEXT!" comment are
50		## taken as text files
51		## - otherwise: exactly the files with names matching the regular expression
52		## ".$\.txt\\|\.gi\\|\.gd\\|\.g\\|\.c\\|\.h\\|\.htm\\|\.html\\|\.xml\\|\.tex\\|\.six\\|\.bib\\|\.tst\\|README.\\|INSTALL.*\\|Makefile$"
53		## are taken as text files
54		##
55		## (Remark: Just providing a .tar.gz file will often result in useful
56		## archives)
57		##
58		## These entries are optional.
59		#TextFiles := ["init.g", ......],
60	27	BinaryFiles := ["doc/manual.dvi", "doc/manual.pdf"],
61	28
62
63		## Information about authors and maintainers. Specify for each person a
64		## record with the following information:
65		##
66		## rec(
67		## # these are compulsory, characters are interpreted as latin-1, so
68		## # German umlauts and other western European special characters are ok:
69		## LastName := "Müller",
70		## FirstNames := "Fritz Eduard",
71		##
72		## # At least one of the following two entries must be given and set
73		## # to 'true' (an entry can be left out if value is not 'true'):
74		## IsAuthor := true;
75		## IsMaintainer := true;
76		##
77		## # At least one of the following three entries must be given.
78		## # - preferably email address and WWW homepage
79		## # - postal address not needed if email or WWW address available
80		## # - if no contact known, specify postal address as "no address known"
81		## Email := "Mueller@no.org",
82		## # complete URL, starting with protocol
83		## WWWHome := "http://www.no.org/~Mueller",
84		## # separate lines by '\n' (optional)
85		## PostalAddress := "Dr. F. Müller\nNo Org Institute\nNo Place 13\n\
86		## 12345 Notown\nNocountry"
87		##
88		## # If you want, add one or both of the following entries (optional)
89		## Place := "Notown",
90		## Institution := "Institute for Nothing"
91		## )
92		##
93	29	Persons := [
94	30	rec(
95	31	LastName := "Soicher",

97	33	IsAuthor := true,
98	34	IsMaintainer := true,
99	35	Email := "L.H.Soicher@qmul.ac.uk",
100		WWWHome := "http://www.maths.qmul.ac.uk/~leonard/",
	36	WWWHome := "http://www.maths.qmul.ac.uk/~lsoicher/",
101	37	Place := "London",
102	38	Institution := Concatenation( [
103	39	"School of Mathematical Sciences, ",
104		"Queen Mary, University of London",
	40	"Queen Mary University of London",
105	41	] )
106	42	)
107		# provide such a record for each author and/or maintainer ...
108
109	43	],
110	44
111	45	## Status information. Currently the following cases are recognized:

124	58	# format: mm/yyyy
125	59	AcceptDate := "08/2006",
126	60
127		## For a central overview of all packages and a collection of all package
128		## archives it is necessary to have two files accessible which should be
129		## contained in each package:
130		## - A README file, containing a short abstract about the package
131		## content and installation instructions.
132		## - The file you are currently reading or editing!
133		## You must specify URLs for these two files, these allow to automate
134		## the updating of package information on the GAP Website, and inclusion
135		## and updating of the package in the GAP distribution.
136		##
137		README_URL := "http://www.maths.qmul.ac.uk/~leonard/design/design1r6/README",
138		PackageInfoURL := "http://www.maths.qmul.ac.uk/~leonard/design/design1r6/PackageInfo.g",
139
140	61	## Here you must provide a short abstract explaining the package content
141	62	## in HTML format (used on the package overview Web page) and an URL
142	63	## for a Webpage with more detailed information about the package

145	66	## '<span class="pkgname">MyPKG</span>' for specifing package names.
146	67	##
147	68	AbstractHTML := "<span class=\"pkgname\">DESIGN</span> is a package for \
148		constructing, classifying, partitioning and studying block designs.",
149
150		PackageWWWHome := "http://www.designtheory.org/software/gap_design/",
	69	constructing, classifying, partitioning, and studying block designs.",
151	70
152	71	## On the GAP Website there is an online version of all manuals in the
153	72	## GAP distribution. To handle the documentation of a package it is

189	108	Dependencies := rec(
190	109	# GAP version, use version strings for specifying exact versions,
191	110	# prepend a '>=' for specifying a least version.
192		GAP := ">=4.5",
	111	GAP := ">=4.10",
193	112	# list of pairs [package name, (least) version], package name is case
194	113	# insensitive, least version denoted with '>=' prepended to version string.
195	114	# without these, the package will not load
196	115	# NeededOtherPackages := [["GAPDoc", ">= 0.99"]],
197		NeededOtherPackages := [["GRAPE", ">= 4.4"]],
	116	NeededOtherPackages := [["GRAPE", ">= 4.8"], ["GAPDoc", ">=1.6"]],
198	117	# without these the package will issue a warning while loading
199	118	# SuggestedOtherPackages := [],
200		SuggestedOtherPackages := [["GAPDoc", ">= 1.4"]],
	119	SuggestedOtherPackages := [],
201	120	# needed external conditions (programs, operating system, ...) provide
202	121	# just strings as text or
203	122	# pairs [text, URL] where URL provides further information

220	139	# AvailabilityTest := ReturnTrue,
221	140	AvailabilityTest := ReturnTrue,
222	141
223		## Suggest here if the package should be automatically loaded when GAP is
224		## started. This should usually be 'false'. Say 'true' only if your package
225		## provides some improvements of the GAP library which are likely to enhance
226		## the overall system performance for many users.
227		Autoload := false,
228
229		## If the default banner does not suffice then provide a string that is
230		## printed when the package is loaded (not when it is autoloaded or if
231		## command line options `-b' or `-q' are given).
232		# BannerString :=
233		# "\nLoading GRAPE 4.2 (GRaph Algorithms using PErmutation groups),\n\
234		# by L.H.Soicher@qmul.ac.uk.\n\n",
235
236		Subtitle := "The Design Package for GAP",
237
238	142	## Optional, but recommended: path relative to package root to a file which
239	143	## contains as many tests of the package functionality as sensible.
240		#TestFile := "tst/testall.g",
	144	TestFile := "tst/testall.g",
241	145
242	146	## Optional: Here you can list some keyword related to the topic
243	147	## of the package.

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~~README~~ less more

0		The DESIGN Package for GAP
1
2		This README describes the {\DESIGN}~1.6 package for {\GAP} (version at
3		least 4.5). The {\DESIGN} package is for constructing, classifying,
4		partitioning and studying block designs.
5
6		All {\DESIGN} functions are written entirely in the {\GAP} language.
7		However, {\DESIGN} requires the {\GRAPE} package (version
8		at least 4.4) to be installed, and makes use of certain {\GRAPE}
9		functions, some of which make use of B.D.~McKay's {\nauty} package.
10		These {\GRAPE} functions can only be used on a fully installed version
11		of {\GRAPE}.
12
13		The {\DESIGN} package is Copyright {\copyright} Leonard H. Soicher
14		2003--2011. {\DESIGN} is part of a wider project, which received EPSRC
15		funding under grant GR/R29659/01, to provide a web-based resource for
16		design theory; see \URL{http://designtheory.org}.
17
18		{\DESIGN} is free software; you can redistribute it and/or modify
19		it under the terms of the GNU General Public License as published by
20		the Free Software Foundation; either version 2 of the License, or
21		(at your option) any later version. For details, see
22		\URL{http://www.gnu.org/licenses/gpl.html}
23
24		If you use {\DESIGN} to solve a problem then please send a short email
25		about it to \Mailto{L.H.Soicher@qmul.ac.uk}, and reference the {\DESIGN}
26		package as follows:
27
28		L.H. Soicher, The DESIGN package for GAP, Version 1.6, 2011,
29		http://designtheory.org/software/gap_design/.
30
31		Any comments or bug reports should also go to
32		\Mailto{L.H.Soicher@qmul.ac.uk}.
33
34		Installing the DESIGN Package
35		-----------------------------
36
37		The {\DESIGN} package is included in the standard {\GAP}~4.5
38		distribution. You only need to download and install {\DESIGN} if you need
39		to install the package locally or are installing an upgrade of {\DESIGN}
40		to an existing installation of {\GAP} (see the main {\GAP} reference
41		section "Reference: Installing a GAP Package"). If you do need to download
42		{\DESIGN}, you can find archive files for the package in various formats
43		at \URL{http://www.gap-system.org/Packages/design.html}, and then your
44		archive file of choice should be downloaded and unpacked in the `pkg'
45		subdirectory of an appropriate GAP root directory (see the main {\GAP}
46		reference section "Reference: GAP Root Directory").
47
48		The {\DESIGN} package is written entirely in {\GAP} code, and requires
49		no further installation. However, the {\DESIGN} package has complete
50		functionality only in an environment in which the {\GRAPE} package is
51		fully installed. Usually this will be on a Unix system.
52
53		Main change from DESIGN 1.5 to DESIGN 1.6
54		-----------------------------------------
55
56		(1) Revised installation instructions.
57
58		Main changes from DESIGN 1.4 to DESIGN 1.5
59		------------------------------------------
60
61		(1) Added internal function:
62		DESIGN_IntervalForLeastRealZero
63
64		(2) Added user functions:
65		PointBlockIncidenceMatrix
66		ConcurrenceMatrix
67		InformationMatrix
68		BlockDesignEfficiency
69
70		(3) Updated documentation, including documentation for
71		the new user functions.
72
73		Main changes from DESIGN 1.3 to DESIGN 1.4
74		------------------------------------------
75
76		(1) Added function: AGPointFlatBlockDesign.
77
78		(2) Improved and documented the main features of function
79		SemiLatinSquareDuals, which can be used to classify (n x n)/k semi-Latin
80		squares and SOMA(k,n)s with given properties.
81
82		(3) Improved efficiency of function PGPointFlatBlockDesign.
83
84		(4) Made function TDesignLambdas faster, in particular in the case of
85		a point-transitive design, when that design's record includes a known
86		point-transitive subgroup of the design's automorphism group.
87
88		(5) Improved the (undocumented) function IsPossiblySBIBD so that
89		TDesignBlockMultiplicityBound now makes full use of the Bruck-Ryser-Chowla
90		Theorem when t=2 and k<v=b.
91
92		(6) Improved the speed of functions TDesignBlockMultiplicityBound and
93		ResolvableTDesignBlockMultiplicityBound by employing binary search to
94		find the bounds.
95
96		Main changes from DESIGN 1.2 to DESIGN 1.3
97		------------------------------------------
98
99		(1) Added function: WittDesign.
100
101		(2) Fixed a bug which caused a break loop to be entered in certain cases
102		in function BlockDesigns when the least element of blockSizes was equal
103		to tSubsetStructure.t.
104
105		(3) Fixed a rarely occurring bug which caused no designs to be returned
106		in function BlockDesigns in certain cases when the designs had just one
107		block each.
108
109		(4) Small improvements made to the documentation and the code.
110
111		Main changes from DESIGN 1.1 to DESIGN 1.2
112		------------------------------------------
113
114		(1) Added functions:
115		BlockIntersectionPolynomial
116		BlockIntersectionPolynomialCheck
117		TDesignLambdas
118		TDesignLambdaMin
119		TDesignIntersectionTriangle
120		SteinerSystemIntersectionTriangle
121		TDesignBlockMultiplicityBound
122		ResolvableTDesignBlockMultiplicityBound
123		BlockDesignPoints
124		NrBlockDesignPoints
125		BlockDesignBlocks
126		NrBlockDesignBlocks
127		PairwiseBalancedLambda
128		AffineResolvableMu
129		DeletedPointsBlockDesign
130		DeletedBlocksBlockDesign
131		AddedPointBlockDesign
132		AddedBlocksBlockDesign
133		DerivedBlockDesign
134		ResidualBlockDesign
135
136		(2) The function BlockDesign can now (optionally) construct a block
137		design from base blocks and a group of automorphisms, and the list
138		of blocks given to BlockDesign need no longer be sorted.
139
140		(3) Improved documentation of the function TDesignFromTBD, and changed
141		the parameters (in a backward compatible way) to match the description
142		of the "*-construction" in the McSorley-Soicher paper.
143
144		(4) The (undocumented) resolveSimple option removed from functions
145		BlockDesigns and PartitionsIntoBlockDesigns, and the option b added
146		to these functions (to specify the number of blocks in a required block
147		design or partition part).
148
149		(5) Improvements made to the use of store (and time?) by the function
150		BlockDesigns.
151
152		Main changes from DESIGN 1.0 to DESIGN 1.1
153		------------------------------------------
154
155		(1) Made compatible with GAP 4.4 and its package loading mechanism.
156		DESIGN 1.1 works only with GAP 4.4/GRAPE 4.2/GAPDoc.
157
158		(2) Upgraded XML I/O to DTRS protocol 2.0.
159
160		(3) BlockDesignsToXMLFile has additional parameters to allow more user
161		control over expansion, output, and design id's. The default now is for
162		minimal expansion and writing out, which is a change from DESIGN 1.0.
163
164		(4) Added functions:
165		BlockDesignIsomorphismClassRepresentatives,
166		DualBlockDesign,
167		ComplementBlocksBlockDesign.
168
169		(5) Function names made read-only.
170
171		(6) Some utility functionality moved to GRAPE 4.2.
172
173		(7) Added files README and gpl.txt.
174
175		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
176		Leonard Soicher
177		Queen Mary, University of London
178		November 2011

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README.md less more

	0	The DESIGN Package for GAP
	1	==========================
	2
	3	The DESIGN package is for constructing, classifying, partitioning,
	4	and studying block designs.
	5
	6	The DESIGN package is Copyright (C) Leonard H. Soicher 2003--2019.
	7
	8	DESIGN is free software; you can redistribute it and/or modify
	9	it under the terms of the GNU General Public License as published by
	10	the Free Software Foundation; either version 2 of the License, or
	11	(at your option) any later version. For details, see
	12	<http://www.gnu.org/licenses/gpl.html>.
	13
	14	Please reference your use of the DESIGN package in a published work
	15	as follows:
	16
	17	L.H. Soicher, The DESIGN package for GAP, Version 1.7, 2019,
	18	<https://gap-packages.github.io/design>.
	19
	20	Any comments or bug reports should go to <L.H.Soicher@qmul.ac.uk>.
	21
	22	Installing the DESIGN Package
	23	-----------------------------
	24
	25	The DESIGN package is included in the standard GAP distribution. You
	26	only need to download and install DESIGN if you need to install
	27	the package locally or are installing an upgrade of DESIGN to an
	28	existing installation of GAP (see the main GAP reference section
	29	"ref:Installing a GAP Package"). If you do need to download DESIGN,
	30	you can find archive files for the package in various formats at
	31	<https://www.gap-system.org/Packages/design.html>, and then your archive
	32	file of choice should be downloaded and unpacked in the `pkg' subdirectory
	33	of an appropriate GAP root directory (see the main GAP reference section
	34	"ref:GAP Root Directories").
	35
	36	The DESIGN package is written entirely in GAP code, and requires
	37	no further installation. However, the DESIGN package has complete
	38	functionality only in an environment in which the GRAPE package is
	39	fully installed.
	40
	41	-------------------------------------------------------------------------
	42	Leonard Soicher,
	43	Queen Mary University of London,
	44	March 2019

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doc/autiso.tex less more

6	6	\def\DESIGN{\sf DESIGN}
7	7	\def\GRAPE{\sf GRAPE}
8	8	\def\nauty{\it nauty}
	9	\def\bliss{\it bliss}
9	10	\def\Aut{{\rm Aut}\,}
10	11
11	12	\Chapter{Automorphism groups and isomorphism testing for block designs}
12	13
13		The functions in this chapter depend on {\nauty} via the {\GRAPE} package,
14		which must be fully installed in order for these functions to work.
	14	The functions in this chapter depend on {\nauty} \cite{Nau90}, \cite{MP14}
	15	or {\bliss} \cite{JK07} via the {\GRAPE} package, which must be fully
	16	installed in order for these functions to work.
15	17
16	18	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
17	19	\Section{Computing automorphism groups}

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doc/construct.tex less more

347	347	\>DerivedBlockDesign( <D>, <x> )
348	348
349	349	Suppose <D> is a block design, and <x> is a point or block of <D>.
350		Then this function returns the derived design\index{derived design}
	350	Then this function returns the derived design
	351	\index{derived design}
351	352	$DD$ of <D>, with respect to <x>.
352	353
353	354	If <x> is a point then $DD$ is the block design whose blocks are those

402	403	\>ResidualBlockDesign( <D>, <x> )
403	404
404	405	Suppose <D> is a block design, and <x> is a point or block of <D>.
405		Then this function returns the residual design\index{residual design}
	406	Then this function returns the residual design
	407	\index{residual design}
406	408	$RD$ of <D>, with respect to <x>.
407	409
408	410	If <x> is a point then $RD$ is the block design whose blocks are those

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doc/design.tex less more

11	11
12	12	\Chapter{Design}
13	13
14		This manual describes the {\DESIGN}~1.6 package for {\GAP} (version at
15		least 4.5). The {\DESIGN} package is for constructing, classifying,
16		partitioning and studying block designs.
17
18		All {\DESIGN} functions are written entirely in the {\GAP} language.
19		However, {\DESIGN} requires the {\GRAPE} package \cite{Grape} (version
20		at least 4.4) to be installed, and makes use of certain {\GRAPE}
21		functions, some of which make use of B.D.~McKay{\pif}s {\nauty} package
22		\cite{Nauty}. These {\GRAPE} functions can only be used on a fully
23		installed version of {\GRAPE}.
	14	This manual describes the {\DESIGN}~1.7 package for {\GAP}. The {\DESIGN}
	15	package is for constructing, classifying, partitioning, and studying
	16	block designs.
24	17
25	18	The {\DESIGN} package is Copyright {\copyright} Leonard H. Soicher
26		2003--2011. {\DESIGN} is part of a wider project, which received EPSRC
	19	2003--2019. {\DESIGN} is part of a wider project, which received EPSRC
27	20	funding under grant GR/R29659/01, to provide a web-based resource for
28	21	design theory; see \URL{http://designtheory.org} and \cite{Dotw}.
29	22

33	26	(at your option) any later version. For details, see
34	27	\URL{http://www.gnu.org/licenses/gpl.html}
35	28
36		If you use {\DESIGN} to solve a problem then please send a short email
37		about it to \Mailto{L.H.Soicher@qmul.ac.uk}, and reference the {\DESIGN}
38		package as follows:
	29	Please reference your use of the {\DESIGN} package in a published work
	30	as follows:
39	31
40		L.H. Soicher, The {DESIGN} package for {GAP}, Version~1.6, 2011,
41		\URL{http://designtheory.org/software/gap_design/}.
	32	L.H.~Soicher, The DESIGN package for GAP, Version 1.7, 2019,
	33	\URL{https://gap-packages.github.io/design}.
42	34
43		Any comments or bug reports should also go to
	35	Any comments or bug reports should go to
44	36	\Mailto{L.H.Soicher@qmul.ac.uk}.
45	37
46	38	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
47	39	\Section{Installing the DESIGN Package}
48	40
49		The {\DESIGN} package is included in the standard {\GAP}~4.5
	41	The {\DESIGN} package is included in the standard {\GAP}
50	42	distribution. You only need to download and install {\DESIGN} if you need
51	43	to install the package locally or are installing an upgrade of {\DESIGN}
52	44	to an existing installation of {\GAP} (see the main {\GAP} reference
53		section "Reference: Installing a GAP Package"). If you do need to download
	45	section "ref:Installing a GAP Package"). If you do need to download
54	46	{\DESIGN}, you can find archive files for the package in various formats
55	47	at \URL{http://www.gap-system.org/Packages/design.html}, and then your
56	48	archive file of choice should be downloaded and unpacked in the `pkg'
57	49	subdirectory of an appropriate GAP root directory (see the main {\GAP}
58		reference section "Reference: GAP Root Directory").
	50	reference section "ref:GAP Root Directories").
59	51
60	52	The {\DESIGN} package is written entirely in {\GAP} code, and requires
61		no further installation. However, the {\DESIGN} package has complete
62		functionality only in an environment in which the {\GRAPE} package is
63		fully installed. Usually this will be on a Unix system.
	53	no further installation. However, {\DESIGN} makes use of the {\GRAPE}
	54	package \cite{Grape}, which must be fully installed.
64	55
65	56	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
66	57	\Section{Loading DESIGN}

69	60	the statement
70	61
71	62	\begintt
72		gap> LoadPackage("design");
73		true
	63	LoadPackage("design");
74	64	\endtt
75	65
76	66	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
77	67	\Section{The structure of a block design in DESIGN}
78	68
79		A block design\index{block design} is an ordered pair $(X,B)$, where
	69	A block design
	70	\index{block design}
	71	is an ordered pair $(X,B)$, where
80	72	$X$ is a non-empty finite set whose elements are called points, and
81	73	$B$ is a non-empty finite multiset whose elements are called blocks,
82	74	such that each block is a non-empty finite multiset of points.
83	75
84	76	{\DESIGN} deals with arbitrary block designs. However, at present, some
85		{\DESIGN} functions only work for binary block designs\index{binary
86		block design} (i.e. those with no repeated element in any block of
	77	{\DESIGN} functions only work for binary block designs
	78	\index{binary block design}
	79	(i.e. those with no repeated element in any block of
87	80	the design), but these functions will check if an input block design
88	81	is binary.
89	82

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doc/efficiency.tex less more

13	13	In this chapter we describe functions to calculate certain matrices
14	14	associated with a block design, and the function `BlockDesignEfficiency'
15	15	which determines certain statistical efficiency measures of a 1-design.
	16
	17	We also document the utility function `DESIGN_IntervalForLeastRealZero',
	18	which is used in the calculation of E-efficiency measures, but has much
	19	wider application.
16	20
17	21	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
18	22	\Section{Matrices associated with a block design}

186	190	Einterval := [ 3/4, 3/4 ], MV := 3/4 )
187	191	\endexample
188	192
189
	193	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	194	\Section{Computing an interval for a certain real zero of a rational polynomial}
	195
	196	We document a {\DESIGN} package utility function used in the calculation
	197	of the `Einterval' component above, but is more widely applicable.
	198
	199	\>DESIGN_IntervalForLeastRealZero( <f>, <a>, <b>, <eps> )
	200
	201	Suppose that <f> is a univariate polynomial over the rationals, <a>,
	202	<b> are rational numbers with $<a>\le <b>$, and <eps> is a positive
	203	rational number.
	204
	205	If <f> has no real zero in the closed interval $[<a>,<b>]$, then this
	206	function returns the empty list. Otherwise, let $\alpha$ be the least
	207	real zero of <f> such that $<a>\le \alpha\le <b>$. Then this function
	208	returns a list $[c,d]$ of rational numbers, with $c\le \alpha\le d$
	209	and $d-c\le <eps>$. Moreover, $c=d$ if and only if $\alpha$ is rational
	210	(in which case $\alpha=c=d$).
	211
	212	\beginexample
	213	gap> x:=Indeterminate(Rationals,1);
	214	x_1
	215	gap> f:=(x+3)*(x^2-3);
	216	x_1^3+3x_1^2-3x_1-9
	217	gap> L:=DESIGN_IntervalForLeastRealZero(f,-5,5,10^(-3));
	218	[ -3, -3 ]
	219	gap> L:=DESIGN_IntervalForLeastRealZero(f,-2,5,10^(-3));
	220	[ -14193/8192, -7093/4096 ]
	221	gap> List(L,Float);
	222	[ -1.73254, -1.73169 ]
	223	gap> L:=DESIGN_IntervalForLeastRealZero(f,0,5,10^(-3));
	224	[ 14185/8192, 7095/4096 ]
	225	gap> List(L,Float);
	226	[ 1.73157, 1.73218 ]
	227	gap> L:=DESIGN_IntervalForLeastRealZero(f,0,5,10^(-5));
	228	[ 454045/262144, 908095/524288 ]
	229	gap> List(L,Float);
	230	[ 1.73204, 1.73205 ]
	231	gap> L:=DESIGN_IntervalForLeastRealZero(f,2,5,10^(-5));
	232	[ ]
	233	\endexample
	234

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doc/make_doc less more

28	28	# The HTML version of the manual
29	29	mkdir -p ../htm
30	30	echo "Creating HTML documentation"
31		../../../etc/convert.pl -c -n design . ../htm
	31	../../../etc/convert.pl -t -c -n design . ../htm
32	32
33	33	#############################################################################
34	34	##

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79	79
80	80	@manual{GAPDoc,
81	81	author = {F. L\"ubeck and M. Neunh\"offer},
82		title = "The GAPDoc package for GAP, Version 1.2",
83		year = "2008",
	82	title = "The GAPDoc package for GAP, Version 1.6.2",
	83	year = "2018",
84	84	note = "\URL{http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/}",
85		}
86
87		@manual{Nauty,
88		author = "B.~D. McKay",
89		title = "{nauty}, Version 2.2",
90		year = "2005",
91		note = "\URL{http://cs.anu.edu.au/people/bdm/nauty/}",
92	85	}
93	86
94	87	@article{CaSo,

124	117
125	118	@manual{Grape,
126	119	author = "L.~H. Soicher",
127		title = "The GRAPE package for GAP, Version 4.5",
128		year = "2011",
129		note = "\URL{http://www.maths.qmul.ac.uk/~leonard/grape/}",
	120	title = "The GRAPE package for GAP, Version 4.8.2",
	121	year = "2019",
	122	note = "\URL{https://gap-packages.github.io/grape}",
130	123	}
131	124
	125	@manual{Nau90,
	126	author = "Brendan D. McKay",
	127	title = "{nauty} user's guide (version 1.5),
	128	Technical report TR-CS-90-02",
	129	year = "1990",
	130	organization = "Australian National University",
	131	address = "Computer Science Department",
	132	note = "nauty homepage:
	133	\URL{http://cs.anu.edu.au/people/bdm/nauty/}",
	134	}
	135
	136	@inproceedings{JK07,
	137	author = "Tommi Juntilla and Petteri Kaski",
	138	title = "Engineering an efficient canonical labeling tool for large and sparse graphs",
	139	crossref = "ALENEX07",
	140	pages = "135--149",
	141	note = "bliss homepage: \URL{http://www.tcs.hut.fi/Software/bliss/}",
	142	}
	143
	144	@proceedings{ALENEX07,
	145	title = "2007 Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments (ALENEX)",
	146	booktitle = "Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments and the
	147	Fourth Workshop on Analytic Algorithmics and Combinatorics",
	148	editor = "David Applegate et al.",
	149	publisher = "SIAM",
	150	year = "2007",
	151	}
	152
	153	@article{MP14,
	154	title = "Practical graph isomorphism, II",
	155	author = "Brendan D. McKay and Adolfo Piperno",
	156	journal = "J. Symbolic Comput.",
	157	volume = "60",
	158	pages = "94--112",
	159	year = "2014",
	160	doi = "http://dx.doi.org/10.1016/j.jsc.2013.09.003",
	161	}
	162

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	0	\makelabel{design:Design}{1}
	1	\makelabel{design:Installing the DESIGN Package}{1.1}
	2	\makelabel{design:Loading DESIGN}{1.2}
	3	\makelabel{design:The structure of a block design in DESIGN}{1.3}
	4	\makelabel{design:Example of the use of DESIGN}{1.4}
	5	\makelabel{design:Information from block design parameters}{2}
	6	\makelabel{design:Information from $t$-design parameters}{2.1}
	7	\makelabel{design:TDesignLambdas}{2.1.1}
	8	\makelabel{design:TDesignLambdaMin}{2.1.2}
	9	\makelabel{design:TDesignIntersectionTriangle}{2.1.3}
	10	\makelabel{design:SteinerSystemIntersectionTriangle}{2.1.4}
	11	\makelabel{design:TDesignBlockMultiplicityBound}{2.1.5}
	12	\makelabel{design:ResolvableTDesignBlockMultiplicityBound}{2.1.6}
	13	\makelabel{design:Block intersection polynomials}{2.2}
	14	\makelabel{design:BlockIntersectionPolynomial}{2.2.1}
	15	\makelabel{design:BlockIntersectionPolynomialCheck}{2.2.2}
	16	\makelabel{design:Constructing block designs}{3}
	17	\makelabel{design:Functions to construct block designs}{3.1}
	18	\makelabel{design:BlockDesign}{3.1.1}
	19	\makelabel{design:BlockDesign}{3.1.1}
	20	\makelabel{design:AGPointFlatBlockDesign}{3.1.2}
	21	\makelabel{design:PGPointFlatBlockDesign}{3.1.3}
	22	\makelabel{design:WittDesign}{3.1.4}
	23	\makelabel{design:DualBlockDesign}{3.1.5}
	24	\makelabel{design:ComplementBlocksBlockDesign}{3.1.6}
	25	\makelabel{design:DeletedPointsBlockDesign}{3.1.7}
	26	\makelabel{design:DeletedBlocksBlockDesign}{3.1.8}
	27	\makelabel{design:AddedPointBlockDesign}{3.1.9}
	28	\makelabel{design:AddedPointBlockDesign}{3.1.9}
	29	\makelabel{design:AddedBlocksBlockDesign}{3.1.10}
	30	\makelabel{design:DerivedBlockDesign}{3.1.11}
	31	\makelabel{design:ResidualBlockDesign}{3.1.12}
	32	\makelabel{design:TDesignFromTBD}{3.1.13}
	33	\makelabel{design:Determining basic properties of block designs}{4}
	34	\makelabel{design:The functions for basic properties}{4.1}
	35	\makelabel{design:IsBlockDesign}{4.1.1}
	36	\makelabel{design:IsBinaryBlockDesign}{4.1.2}
	37	\makelabel{design:IsSimpleBlockDesign}{4.1.3}
	38	\makelabel{design:IsConnectedBlockDesign}{4.1.4}
	39	\makelabel{design:BlockDesignPoints}{4.1.5}
	40	\makelabel{design:NrBlockDesignPoints}{4.1.6}
	41	\makelabel{design:BlockDesignBlocks}{4.1.7}
	42	\makelabel{design:NrBlockDesignBlocks}{4.1.8}
	43	\makelabel{design:BlockSizes}{4.1.9}
	44	\makelabel{design:BlockNumbers}{4.1.10}
	45	\makelabel{design:ReplicationNumber}{4.1.11}
	46	\makelabel{design:PairwiseBalancedLambda}{4.1.12}
	47	\makelabel{design:TSubsetLambdasVector}{4.1.13}
	48	\makelabel{design:AllTDesignLambdas}{4.1.14}
	49	\makelabel{design:AffineResolvableMu}{4.1.15}
	50	\makelabel{design:Matrices and efficiency measures for block designs}{5}
	51	\makelabel{design:Matrices associated with a block design}{5.1}
	52	\makelabel{design:PointBlockIncidenceMatrix}{5.1.1}
	53	\makelabel{design:ConcurrenceMatrix}{5.1.2}
	54	\makelabel{design:InformationMatrix}{5.1.3}
	55	\makelabel{design:The function BlockDesignEfficiency}{5.2}
	56	\makelabel{design:BlockDesignEfficiency}{5.2.1}
	57	\makelabel{design:BlockDesignEfficiency}{5.2.1}
	58	\makelabel{design:BlockDesignEfficiency}{5.2.1}
	59	\makelabel{design:Computing an interval for a certain real zero of a rational polynomial}{5.3}
	60	\makelabel{design:DESIGN_IntervalForLeastRealZero}{5.3.1}
	61	\makelabel{design:Automorphism groups and isomorphism testing for block designs}{6}
	62	\makelabel{design:Computing automorphism groups}{6.1}
	63	\makelabel{design:AutGroupBlockDesign}{6.1.1}
	64	\makelabel{design:Testing isomorphism}{6.2}
	65	\makelabel{design:IsIsomorphicBlockDesign}{6.2.1}
	66	\makelabel{design:BlockDesignIsomorphismClassRepresentatives}{6.2.2}
	67	\makelabel{design:Classifying block designs}{7}
	68	\makelabel{design:The function BlockDesigns}{7.1}
	69	\makelabel{design:BlockDesigns}{7.1.1}
	70	\makelabel{design:Classifying semi-Latin squares}{8}
	71	\makelabel{design:Semi-Latin squares and SOMAs}{8.1}
	72	\makelabel{design:The function SemiLatinSquareDuals}{8.2}
	73	\makelabel{design:SemiLatinSquareDuals}{8.2.1}
	74	\makelabel{design:SemiLatinSquareDuals}{8.2.1}
	75	\makelabel{design:SemiLatinSquareDuals}{8.2.1}
	76	\makelabel{design:SemiLatinSquareDuals}{8.2.1}
	77	\makelabel{design:Partitioning block designs}{9}
	78	\makelabel{design:Partitioning a block design into block designs}{9.1}
	79	\makelabel{design:PartitionsIntoBlockDesigns}{9.1.1}
	80	\makelabel{design:Computing resolutions}{9.2}
	81	\makelabel{design:MakeResolutionsComponent}{9.2.1}
	82	\makelabel{design:MakeResolutionsComponent}{9.2.1}
	83	\makelabel{design:XML I/O of block designs}{10}
	84	\makelabel{design:Writing lists of block designs and their properties in XML-format}{10.1}
	85	\makelabel{design:BlockDesignsToXMLFile}{10.1.1}
	86	\makelabel{design:BlockDesignsToXMLFile}{10.1.1}
	87	\makelabel{design:BlockDesignsToXMLFile}{10.1.1}
	88	\makelabel{design:Reading lists of block designs in XML-format}{10.2}
	89	\makelabel{design:BlockDesignsFromXMLFile}{10.2.1}
	90	\makelabel{design:Bibliography}{}
	91	\setcitlab {BaCh}{BC97}
	92	\setcitlab {BaCa}{BC09}
	93	\setcitlab {Dotw}{BCD{+}06}
	94	\setcitlab {BaRo}{BR97}
	95	\setcitlab {Extrep}{CDMS04}
	96	\setcitlab {CaSo}{CS07}
	97	\setcitlab {JK07}{JK07}
	98	\setcitlab {Nau90}{McK90}
	99	\setcitlab {MP14}{MP14}
	100	\setcitlab {McSo}{MS07}
	101	\setcitlab {Soi1}{Soi10}
	102	\setcitlab {Grape}{Soi19}
	103	\makelabel{design:Index}{}

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	0	C design.tex 1. Design
	1	S 1.1. Installing the DESIGN Package
	2	S 1.2. Loading DESIGN
	3	S 1.3. The structure of a block design in DESIGN
	4	I 1.3. block design
	5	I 1.3. binary block design
	6	S 1.4. Example of the use of DESIGN
	7	C parameters.tex 2. Information from block design parameters
	8	S 2.1. Information from $t$-design parameters
	9	I 2.1. t-design
	10	F 2.1. TDesignLambdas
	11	F 2.1. TDesignLambdaMin
	12	F 2.1. TDesignIntersectionTriangle
	13	F 2.1. SteinerSystemIntersectionTriangle
	14	F 2.1. TDesignBlockMultiplicityBound
	15	F 2.1. ResolvableTDesignBlockMultiplicityBound
	16	S 2.2. Block intersection polynomials
	17	F 2.2. BlockIntersectionPolynomial
	18	F 2.2. BlockIntersectionPolynomialCheck
	19	C construct.tex 3. Constructing block designs
	20	S 3.1. Functions to construct block designs
	21	F 3.1. BlockDesign
	22	F 3.1. BlockDesign
	23	F 3.1. AGPointFlatBlockDesign
	24	F 3.1. PGPointFlatBlockDesign
	25	F 3.1. WittDesign
	26	F 3.1. DualBlockDesign
	27	F 3.1. ComplementBlocksBlockDesign
	28	F 3.1. DeletedPointsBlockDesign
	29	F 3.1. DeletedBlocksBlockDesign
	30	F 3.1. AddedPointBlockDesign
	31	F 3.1. AddedPointBlockDesign
	32	F 3.1. AddedBlocksBlockDesign
	33	F 3.1. DerivedBlockDesign
	34	I 3.1. derived design
	35	F 3.1. ResidualBlockDesign
	36	I 3.1. residual design
	37	F 3.1. TDesignFromTBD
	38	C basic.tex 4. Determining basic properties of block designs
	39	S 4.1. The functions for basic properties
	40	F 4.1. IsBlockDesign
	41	F 4.1. IsBinaryBlockDesign
	42	F 4.1. IsSimpleBlockDesign
	43	F 4.1. IsConnectedBlockDesign
	44	F 4.1. BlockDesignPoints
	45	F 4.1. NrBlockDesignPoints
	46	F 4.1. BlockDesignBlocks
	47	F 4.1. NrBlockDesignBlocks
	48	F 4.1. BlockSizes
	49	F 4.1. BlockNumbers
	50	F 4.1. ReplicationNumber
	51	F 4.1. PairwiseBalancedLambda
	52	F 4.1. TSubsetLambdasVector
	53	F 4.1. AllTDesignLambdas
	54	F 4.1. AffineResolvableMu
	55	C efficiency.tex 5. Matrices and efficiency measures for block designs
	56	S 5.1. Matrices associated with a block design
	57	F 5.1. PointBlockIncidenceMatrix
	58	F 5.1. ConcurrenceMatrix
	59	F 5.1. InformationMatrix
	60	S 5.2. The function BlockDesignEfficiency
	61	F 5.2. BlockDesignEfficiency
	62	F 5.2. BlockDesignEfficiency
	63	F 5.2. BlockDesignEfficiency
	64	S 5.3. Computing an interval for a certain real zero of a rational polynomial
	65	F 5.3. DESIGN_IntervalForLeastRealZero
	66	C autiso.tex 6. Automorphism groups and isomorphism testing for block designs
	67	S 6.1. Computing automorphism groups
	68	F 6.1. AutGroupBlockDesign
	69	S 6.2. Testing isomorphism
	70	F 6.2. IsIsomorphicBlockDesign
	71	F 6.2. BlockDesignIsomorphismClassRepresentatives
	72	C classify.tex 7. Classifying block designs
	73	S 7.1. The function BlockDesigns
	74	F 7.1. BlockDesigns
	75	C sls.tex 8. Classifying semi-Latin squares
	76	S 8.1. Semi-Latin squares and SOMAs
	77	I 8.1. semi-Latin square
	78	I 8.1. SOMA
	79	S 8.2. The function SemiLatinSquareDuals
	80	F 8.2. SemiLatinSquareDuals
	81	F 8.2. SemiLatinSquareDuals
	82	F 8.2. SemiLatinSquareDuals
	83	F 8.2. SemiLatinSquareDuals
	84	C partition.tex 9. Partitioning block designs
	85	S 9.1. Partitioning a block design into block designs
	86	F 9.1. PartitionsIntoBlockDesigns
	87	S 9.2. Computing resolutions
	88	F 9.2. MakeResolutionsComponent
	89	F 9.2. MakeResolutionsComponent
	90	C xmlio.tex 10. XML I/O of block designs
	91	S 10.1. Writing lists of block designs and their properties in XML-format
	92	F 10.1. BlockDesignsToXMLFile
	93	F 10.1. BlockDesignsToXMLFile
	94	F 10.1. BlockDesignsToXMLFile
	95	S 10.2. Reading lists of block designs in XML-format
	96	F 10.2. BlockDesignsFromXMLFile

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22	22	%F UseReferences . . . . . . . . . . . . . . . . . . specify references
23	23	%%
24	24	% \UseReferences{../../../doc/ext}
25		% \UseReferences{../../../doc/ref}
	25	\UseReferences{../../../doc/ref}
26	26	%
27	27	%
28	28	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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doc/parameters.tex less more

14	14	\Section{Information from $t$-design parameters}
15	15
16	16	For $t$ a non-negative integer and $v,k,\lambda$ positive integers
17		with $t\le k\le v$, a $t$-design\index{t-design} with parameters
	17	with $t\le k\le v$, a $t$-design
	18	\index{t-design}
	19	with parameters
18	20	$t,v,k,\lambda$, or a $t$-$(v,k,\lambda)$ design, is a binary block
19	21	design with exactly $v$ points, such that each block has size $k$ and
20	22	each $t$-subset of the points is contained in exactly $\lambda$ blocks.

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doc/sls.tex less more

18	18	\Section{Semi-Latin squares and SOMAs}
19	19
20	20	Let $n$ and $k$ be positive integers. An $(n\times n)/k$ *semi-Latin
21		square*\index{semi-Latin square} is an $n$ by $n$ array $A$, whose
	21	square*
	22	\index{semi-Latin square}
	23	is an $n$ by $n$ array $A$, whose
22	24	entries are $k$-subsets of a $kn$-set $X$ (the symbol-set), such that
23	25	each element of $X$ occurs exactly once in each row and exactly once in
24	26	each column of $A$. (Thus an $(n\times n)/1$ semi-Latin square is the same
25	27	thing as a Latin square of order $n$.) For extensive useful information on
26	28	semi-Latin squares, see \URL{http://www.maths.qmul.ac.uk/~rab/sls.html}.
27	29
28		A SOMA$(k,n)$\index{SOMA} is an $(n\times n)/k$ semi-Latin square $A$,
	30	A SOMA$(k,n)$
	31	\index{SOMA}
	32	is an $(n\times n)/k$ semi-Latin square $A$,
29	33	with $n\ge2$, in which no 2-subset of the symbol-set is contained in
30	34	more than one entry of $A$. For extensive useful information on SOMAs,
31		see \URL{http://www.maths.qmul.ac.uk/~leonard/soma/}.
	35	see \URL{http://www.maths.qmul.ac.uk/~lsoicher/soma/}.
32	36
33	37	Let $A$ and $B$ be $(n\times n)/k$ semi-Latin squares. We say that
34	38	$B$ is (weakly) isomorphic to $A$ if $B$ can be obtained from $A$

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doc/xmlio.tex less more

64	64
65	65	\beginexample
66	66	gap> D:=[ BlockDesign(3, [[1,2],[1,3]]),
67		> BlockDesign(3, [[1,2],[1,2],[2,3]]) ];;
	67	> BlockDesign(3, [[1,2],[1,2],[2,3]]) ];;
68	68	gap> designs:=rec(list:=D, pairwiseNonisomorphic:=true);;
69	69	gap> BlockDesignsToXMLFile("example.xml",designs,[],"example");
70	70	\endexample

100	100	\beginexample
101	101	gap> BlockDesignsFromXMLFile("example.xml");
102	102	rec(
103		infoXML := "<info>\n<software>\n[ DESIGN-1.6, GRAPE-4.5, GAPDoc-1.4, GAP-4.5\
104		.2(beta) ]\n</software>\n</info>",
	103	infoXML := "<info>\n<software>\n[ DESIGN-1.7, GRAPE-4.8.2, GAPDoc-1.6.2, GAP\
	104	-4.10.1 ]\n</software>\n</info>",
105	105	list :=
106	106	[
107	107	rec( blocks := [ [ 1, 2 ], [ 1, 3 ] ], id := "example-0",

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~~gpl.txt~~ less more

0		GNU GENERAL PUBLIC LICENSE
1		Version 2, June 1991
2
3		Copyright (C) 1989, 1991 Free Software Foundation, Inc.
4		59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
5		Everyone is permitted to copy and distribute verbatim copies
6		of this license document, but changing it is not allowed.
7
8		Preamble
9
10		The licenses for most software are designed to take away your
11		freedom to share and change it. By contrast, the GNU General Public
12		License is intended to guarantee your freedom to share and change free
13		software--to make sure the software is free for all its users. This
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15		Foundation's software and to any other program whose authors commit to
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32		For example, if you distribute copies of such a program, whether
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	0	<html><head><title>[design] 1 Design</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
	3	<h1>1 Design</h1><p>
	4	<P>
	5	<H3>Sections</H3>
	6	<oL>
	7	<li> <A HREF="CHAP001.htm#SECT001">Installing the DESIGN Package</a>
	8	<li> <A HREF="CHAP001.htm#SECT002">Loading DESIGN</a>
	9	<li> <A HREF="CHAP001.htm#SECT003">The structure of a block design in DESIGN</a>
	10	<li> <A HREF="CHAP001.htm#SECT004">Example of the use of DESIGN</a>
	11	</ol><p>
	12	<p>
	13	This manual describes the DESIGN 1.7 package for <font face="Gill Sans,Helvetica,Arial">GAP</font>. The DESIGN
	14	package is for constructing, classifying, partitioning, and studying
	15	block designs.
	16	<p>
	17	The DESIGN package is Copyright © Leonard H. Soicher
	18	2003--2019. DESIGN is part of a wider project, which received EPSRC
	19	funding under grant GR/R29659/01, to provide a web-based resource for
	20	design theory; see <a href="http://designtheory.org">http://designtheory.org</a> and <a href="biblio.htm#Dotw"><cite>Dotw</cite></a>.
	21	<p>
	22	DESIGN is free software; you can redistribute it and/or modify
	23	it under the terms of the GNU General Public License as published by
	24	the Free Software Foundation; either version 2 of the License, or
	25	(at your option) any later version. For details, see
	26	<a href="http://www.gnu.org/licenses/gpl.html">http://www.gnu.org/licenses/gpl.html</a>
	27	<p>
	28	Please reference your use of the DESIGN package in a published work
	29	as follows:
	30	<p>
	31	L.H. Soicher, The DESIGN package for GAP, Version 1.7, 2019,
	32	<a href="https://gap-packages.github.io/design">https://gap-packages.github.io/design</a>.
	33	<p>
	34	Any comments or bug reports should go to
	35	<a href="mailto:L.H.Soicher@qmul.ac.uk">L.H.Soicher@qmul.ac.uk</a>.
	36	<p>
	37	<p>
	38	<h2><a name="SECT001">1.1 Installing the DESIGN Package</a></h2>
	39	<p><p>
	40	The DESIGN package is included in the standard <font face="Gill Sans,Helvetica,Arial">GAP</font>
	41	distribution. You only need to download and install DESIGN if you need
	42	to install the package locally or are installing an upgrade of DESIGN
	43	to an existing installation of <font face="Gill Sans,Helvetica,Arial">GAP</font> (see the main <font face="Gill Sans,Helvetica,Arial">GAP</font> reference
	44	section <a href="../../../doc/ref/chap76.html#X82473E4B8756C6CD">Installing a GAP Package</a>). If you do need to download
	45	DESIGN, you can find archive files for the package in various formats
	46	at <a href="http://www.gap-system.org/Packages/design.html">http://www.gap-system.org/Packages/design.html</a>, and then your
	47	archive file of choice should be downloaded and unpacked in the <code>pkg</code>
	48	subdirectory of an appropriate GAP root directory (see the main <font face="Gill Sans,Helvetica,Arial">GAP</font>
	49	reference section <a href="../../../doc/ref/chap9.html#X7A4973627A5DB27D">GAP Root Directories</a>).
	50	<p>
	51	The DESIGN package is written entirely in <font face="Gill Sans,Helvetica,Arial">GAP</font> code, and requires
	52	no further installation. However, DESIGN makes use of the GRAPE
	53	package <a href="biblio.htm#Grape"><cite>Grape</cite></a>, which must be fully installed.
	54	<p>
	55	<p>
	56	<h2><a name="SECT002">1.2 Loading DESIGN</a></h2>
	57	<p><p>
	58	Before using DESIGN you must load the package within <font face="Gill Sans,Helvetica,Arial">GAP</font> by calling
	59	the statement
	60	<p>
	61	<pre>
	62	LoadPackage("design");
	63	</pre>
	64	<p>
	65	<p>
	66	<h2><a name="SECT003">1.3 The structure of a block design in DESIGN</a></h2>
	67	<p><p>
	68	A <strong>block design</strong>
	69	<a name = "I0"></a>
	70
	71	is an ordered pair (<i>X</i>,<i>B</i>), where
	72	<i>X</i> is a non-empty finite set whose elements are called <strong>points</strong>, and
	73	<i>B</i> is a non-empty finite multiset whose elements are called <strong>blocks</strong>,
	74	such that each block is a non-empty finite multiset of points.
	75	<p>
	76	DESIGN deals with arbitrary block designs. However, at present, some
	77	DESIGN functions only work for <strong>binary</strong> block designs
	78	<a name = "I1"></a>
	79
	80	(i.e. those with no repeated element in any block of
	81	the design), but these functions will check if an input block design
	82	is binary.
	83	<p>
	84	In DESIGN, a block design <var>D</var> is stored as a record, with mandatory
	85	components <code>isBlockDesign</code>, <code>v</code>, and <code>blocks</code>. The points of a block
	86	design <var>D</var> are always 1,2,...,<code></code><var>D</var><code>.v</code>, but they may also be given <strong>names</strong>
	87	in the optional component <code>pointNames</code>, with <code></code><var>D</var><code>.pointNames[</code><var>i</var><code>]</code>
	88	the name of point <var>i</var>. The <code>blocks</code> component must be a sorted list
	89	of the blocks of <var>D</var> (including any repeats), with each block being a
	90	sorted list of points (including any repeats).
	91	<p>
	92	A block design record may also have some optional components which store
	93	information about the design. At present these optional components include
	94	<code>isSimple</code>, <code>isBinary</code>, <code>isConnected</code>, <code>r</code>, <code>blockSizes</code>, <code>blockNumbers</code>,
	95	<code>resolutions</code>, <code>autGroup</code>, <code>autSubgroup</code>, <code>tSubsetStructure</code>,
	96	<code>allTDesignLambdas</code>, <code>efficiency</code>, <code>id</code>, <code>statistical_propertiesXML</code>,
	97	and <code>pointNames</code>.
	98	<p>
	99	A non-expert user should only use functions in the DESIGN package to
	100	create block design records and their components.
	101	<p>
	102	<p>
	103	<h2><a name="SECT004">1.4 Example of the use of DESIGN</a></h2>
	104	<p><p>
	105	To give you an idea of the capabilities of this package, we now give
	106	an extended example of an application of the DESIGN package, in
	107	which a nearly resolvable non-simple 2-(21,4,3) design is constructed
	108	(for Donald Preece) via a pairwise-balanced design. All the DESIGN
	109	functions used here are described in this manual.
	110	<p>
	111	The program first discovers the unique (up to isomorphism)
	112	pairwise-balanced 2-(21,{4,5},1) design <i>D</i> invariant under <i>H</i>=〈(1,2,…,20)〉, and then applies the *-construction of
	113	<a href="biblio.htm#McSo"><cite>McSo</cite></a> to this design <i>D</i> to obtain a non-simple 2-(21,4,3) design
	114	<i>Dstar</i> with automorphism group of order 80. The program then classifies
	115	the near-resolutions of <i>Dstar</i> invariant under the subgroup of order 5
	116	of <i>H</i>, and finds exactly two such (up to the action of \Aut(<i>Dstar</i>)).
	117	Finally, <i>Dstar</i> is printed.
	118	<p>
	119	<pre>
	120	gap> H:=CyclicGroup(IsPermGroup,20);
	121	Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20) ])
	122	gap> D:=BlockDesigns(rec(v:=21,blockSizes:=[4,5],
	123	> tSubsetStructure:=rec(t:=2,lambdas:=[1]),
	124	> requiredAutSubgroup:=H ));;
	125	gap> Length(D);
	126	1
	127	gap> D:=D[1];;
	128	gap> BlockSizes(D);
	129	[ 4, 5 ]
	130	gap> BlockNumbers(D);
	131	[ 20, 9 ]
	132	gap> Size(AutGroupBlockDesign(D));
	133	80
	134	gap> Dstar:=TDesignFromTBD(D,2,4);;
	135	gap> AllTDesignLambdas(Dstar);
	136	[ 105, 20, 3 ]
	137	gap> IsSimpleBlockDesign(Dstar);
	138	false
	139	gap> Size(AutGroupBlockDesign(Dstar));
	140	80
	141	gap> near_resolutions:=PartitionsIntoBlockDesigns(rec(
	142	> blockDesign:=Dstar,
	143	> v:=21,blockSizes:=[4],
	144	> tSubsetStructure:=rec(t:=0,lambdas:=[5]),
	145	> blockIntersectionNumbers:=[[ [0] ]],
	146	> requiredAutSubgroup:=SylowSubgroup(H,5) ));;
	147	gap> Length(near_resolutions);
	148	2
	149	gap> List(near_resolutions,x->Size(x.autGroup));
	150	[ 5, 20 ]
	151	gap> Print(Dstar,"\n");
	152	rec(
	153	isBlockDesign := true,
	154	v := 21,
	155	blocks := [ [ 1, 2, 4, 15 ], [ 1, 2, 4, 15 ], [ 1, 2, 4, 15 ],
	156	[ 1, 3, 14, 20 ], [ 1, 3, 14, 20 ], [ 1, 3, 14, 20 ], [ 1, 5, 9, 13 ],
	157	[ 1, 5, 9, 17 ], [ 1, 5, 13, 17 ], [ 1, 6, 11, 16 ], [ 1, 6, 11, 21 ],
	158	[ 1, 6, 16, 21 ], [ 1, 7, 8, 10 ], [ 1, 7, 8, 10 ], [ 1, 7, 8, 10 ],
	159	[ 1, 9, 13, 17 ], [ 1, 11, 16, 21 ], [ 1, 12, 18, 19 ],
	160	[ 1, 12, 18, 19 ], [ 1, 12, 18, 19 ], [ 2, 3, 5, 16 ], [ 2, 3, 5, 16 ],
	161	[ 2, 3, 5, 16 ], [ 2, 6, 10, 14 ], [ 2, 6, 10, 18 ], [ 2, 6, 14, 18 ],
	162	[ 2, 7, 12, 17 ], [ 2, 7, 12, 21 ], [ 2, 7, 17, 21 ], [ 2, 8, 9, 11 ],
	163	[ 2, 8, 9, 11 ], [ 2, 8, 9, 11 ], [ 2, 10, 14, 18 ], [ 2, 12, 17, 21 ],
	164	[ 2, 13, 19, 20 ], [ 2, 13, 19, 20 ], [ 2, 13, 19, 20 ],
	165	[ 3, 4, 6, 17 ], [ 3, 4, 6, 17 ], [ 3, 4, 6, 17 ], [ 3, 7, 11, 15 ],
	166	[ 3, 7, 11, 19 ], [ 3, 7, 15, 19 ], [ 3, 8, 13, 18 ], [ 3, 8, 13, 21 ],
	167	[ 3, 8, 18, 21 ], [ 3, 9, 10, 12 ], [ 3, 9, 10, 12 ], [ 3, 9, 10, 12 ],
	168	[ 3, 11, 15, 19 ], [ 3, 13, 18, 21 ], [ 4, 5, 7, 18 ], [ 4, 5, 7, 18 ],
	169	[ 4, 5, 7, 18 ], [ 4, 8, 12, 16 ], [ 4, 8, 12, 20 ], [ 4, 8, 16, 20 ],
	170	[ 4, 9, 14, 19 ], [ 4, 9, 14, 21 ], [ 4, 9, 19, 21 ], [ 4, 10, 11, 13 ],
	171	[ 4, 10, 11, 13 ], [ 4, 10, 11, 13 ], [ 4, 12, 16, 20 ],
	172	[ 4, 14, 19, 21 ], [ 5, 6, 8, 19 ], [ 5, 6, 8, 19 ], [ 5, 6, 8, 19 ],
	173	[ 5, 9, 13, 17 ], [ 5, 10, 15, 20 ], [ 5, 10, 15, 21 ],
	174	[ 5, 10, 20, 21 ], [ 5, 11, 12, 14 ], [ 5, 11, 12, 14 ],
	175	[ 5, 11, 12, 14 ], [ 5, 15, 20, 21 ], [ 6, 7, 9, 20 ], [ 6, 7, 9, 20 ],
	176	[ 6, 7, 9, 20 ], [ 6, 10, 14, 18 ], [ 6, 11, 16, 21 ],
	177	[ 6, 12, 13, 15 ], [ 6, 12, 13, 15 ], [ 6, 12, 13, 15 ],
	178	[ 7, 11, 15, 19 ], [ 7, 12, 17, 21 ], [ 7, 13, 14, 16 ],
	179	[ 7, 13, 14, 16 ], [ 7, 13, 14, 16 ], [ 8, 12, 16, 20 ],
	180	[ 8, 13, 18, 21 ], [ 8, 14, 15, 17 ], [ 8, 14, 15, 17 ],
	181	[ 8, 14, 15, 17 ], [ 9, 14, 19, 21 ], [ 9, 15, 16, 18 ],
	182	[ 9, 15, 16, 18 ], [ 9, 15, 16, 18 ], [ 10, 15, 20, 21 ],
	183	[ 10, 16, 17, 19 ], [ 10, 16, 17, 19 ], [ 10, 16, 17, 19 ],
	184	[ 11, 17, 18, 20 ], [ 11, 17, 18, 20 ], [ 11, 17, 18, 20 ] ],
	185	autGroup := Group( [ ( 2,14,10,18)( 3, 7,19,15)( 4,20, 8,12)( 5,13,17, 9),
	186	( 1,17, 5, 9)( 2,10,14, 6)( 4,16,12,20)( 7,15,19,11),
	187	( 1,18,19,12)( 2,11, 8, 9)( 3, 4,17, 6)( 5,10,15,20)( 7,16,13,14) ] ),
	188	blockSizes := [ 4 ],
	189	isBinary := true,
	190	allTDesignLambdas := [ 105, 20, 3 ],
	191	isSimple := false )
	192	</pre>
	193	<p>
	194	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
	195	<P>
	196	<address>design manual<br>March 2019
	197	</address></body></html>⏎

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	0	<html><head><title>[design] 2 Information from block design parameters</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP001.htm">Previous</a>] [<a href ="CHAP003.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
	3	<h1>2 Information from block design parameters</h1><p>
	4	<P>
	5	<H3>Sections</H3>
	6	<oL>
	7	<li> <A HREF="CHAP002.htm#SECT001">Information from $t$-design parameters</a>
	8	<li> <A HREF="CHAP002.htm#SECT002">Block intersection polynomials</a>
	9	</ol><p>
	10	<p>
	11	<p>
	12	<h2><a name="SECT001">2.1 Information from $t$-design parameters</a></h2>
	13	<p><p>
	14	For <i>t</i> a non-negative integer and <i>v</i>,<i>k</i>,λ positive integers
	15	with <i>t</i> ≤ <i>k</i> ≤ <i>v</i>, a <i>t</i>-<strong>design</strong>
	16	<a name = "I0"></a>
	17
	18	with <strong>parameters</strong>
	19	<i>t</i>,<i>v</i>,<i>k</i>,λ, or a <i>t</i>-(<i>v</i>,<i>k</i>,λ) <strong>design</strong>, is a binary block
	20	design with exactly <i>v</i> points, such that each block has size <i>k</i> and
	21	each <i>t</i>-subset of the points is contained in exactly λ blocks.
	22	<p>
	23	<a name = "SSEC001.1"></a>
	24	<li><code>TDesignLambdas( </code><var>t</var><code>, </code><var>v</var><code>, </code><var>k</var><code>, </code><var>lambda</var><code> )</code>
	25	<p>
	26	A <i>t</i>-(<i>v</i>,<i>k</i>,λ) design is also an <i>s</i>-(<i>v</i>,<i>k</i>,λ<sub><i>s</i></sub>) design
	27	for 0 ≤ <i>s</i> ≤ <i>t</i>, where <br clear="all" /><table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center">λ<sub><i>s</i></sub>=λ</td><td align="left" class="cl">⎛<br />⎝</td><td nowrap="nowrap" align="center"><i>v</i>−<i>s</i><br /><i>t</i>−<i>s</i><br /></td><td align="left" class="cl">⎞<br />⎠</td><td nowrap="nowrap" align="center">/</td><td align="left" class="cl">⎛<br />⎝</td><td nowrap="nowrap" align="center"><i>k</i>−<i>s</i><br /><i>t</i>−<i>s</i><br /></td><td align="left" class="cl">⎞<br />⎠</td><td nowrap="nowrap" align="center"></td></tr></table></td></tr></table>.
	28	<p>
	29	Given a non-negative integer <var>t</var>, and positive integers <var>v</var>, <var>k</var>,
	30	<var>lambda</var>, with <i>t</i> ≤ <i>k</i> ≤ <i>v</i> , this function returns a length
	31	<i>t</i> +1 list whose (<i>s</i>+1)-st element is λ<sub><i>s</i></sub> as defined above,
	32	if all the λ<sub><i>s</i></sub> are integers. Otherwise, <code>fail</code> is returned.
	33	<p>
	34	<pre>
	35	gap> TDesignLambdas(5,24,8,1);
	36	[ 759, 253, 77, 21, 5, 1 ]
	37	</pre>
	38	<p>
	39	<a name = "SSEC001.2"></a>
	40	<li><code>TDesignLambdaMin( </code><var>t</var><code>, </code><var>v</var><code>, </code><var>k</var><code> )</code>
	41	<p>
	42	Given a non-negative integer <var>t</var>, and positive integers <var>v</var> and <var>k</var>, with
	43	<i>t</i> ≤ <i>k</i> ≤ <i>v</i> , this function returns the minimum positive <var>lambda</var>
	44	such that <code>TDesignLambdas( </code><var>t</var><code>, </code><var>v</var><code>, </code><var>k</var><code>, </code><var>lambda</var><code> )</code> does not return
	45	<code>fail</code>.
	46	<p>
	47	See <a href="CHAP002.htm#SSEC001.1">TDesignLambdas</a>.
	48	<p>
	49	<pre>
	50	gap> TDesignLambdaMin(5,24,8);
	51	1
	52	gap> TDesignLambdaMin(2,12,4);
	53	3
	54	</pre>
	55	<p>
	56	<a name = "SSEC001.3"></a>
	57	<li><code>TDesignIntersectionTriangle( </code><var>t</var><code>, </code><var>v</var><code>, </code><var>k</var><code>, </code><var>lambda</var><code> )</code>
	58	<p>
	59	Suppose <i>D</i> is a <var>t</var>-(<var>v</var>,<var>k</var>,<var>lambda</var>) design, let <i>i</i> and <i>j</i>
	60	be non-negative integers with <i>i</i>+<i>j</i> ≤ <i>t</i>, and suppose <i>X</i> and <i>Y</i>
	61	are disjoint subsets of the points of <i>D</i>, such that <i>X</i> and <i>Y</i> have
	62	respective sizes <i>i</i> and <i>j</i>. The (<i>i</i>,<i>j</i>)-<strong>intersection number</strong> is
	63	the number of blocks of <i>D</i> that contain <i>X</i> and are disjoint from <i>Y</i>
	64	(this number depends only on <var>t</var>, <var>v</var>, <var>k</var>, <var>lambda</var>, <i>i</i> and <i>j</i>).
	65	<p>
	66	Given a non-negative integer <var>t</var>, and positive integers <var>v</var>, <var>k</var>
	67	and <var>lambda</var>, with <i>t</i> ≤ <i>k</i> ≤ <i>v</i> , this function returns the
	68	<strong><var>t</var>-design intersection triangle</strong>, which is a two dimensional array
	69	whose (<i>i</i>+1,<i>j</i>+1)-entry is the (<i>i</i>,<i>j</i>)-intersection number for
	70	a <var>t</var>-(<var>v</var>,<var>k</var>,<var>lambda</var>) design (assuming such a design exists),
	71	such that <i>i</i>,<i>j</i> ≥ 0, <i>i</i>+<i>j</i> ≤ <i>t</i>. This function returns <code>fail</code> if
	72	<code>TDesignLambdas(</code><var>t</var><code>,</code><var>v</var><code>,</code><var>k</var><code>,</code><var>lambda</var><code>)</code> does. When <i>lambda</i> =1, then more
	73	information can be obtained using <a href="CHAP002.htm#SSEC001.4">SteinerSystemIntersectionTriangle</a>.
	74	<p>
	75	<pre>
	76	gap> TDesignLambdas(2,12,4,3);
	77	[ 33, 11, 3 ]
	78	gap> TDesignIntersectionTriangle(2,12,4,3);
	79	[ [ 33, 22, 14 ], [ 11, 8 ], [ 3 ] ]
	80	gap> TDesignLambdas(2,12,4,2);
	81	fail
	82	gap> TDesignIntersectionTriangle(2,12,4,2);
	83	fail
	84	</pre>
	85	<p>
	86	<a name = "SSEC001.4"></a>
	87	<li><code>SteinerSystemIntersectionTriangle( </code><var>t</var><code>, </code><var>v</var><code>, </code><var>k</var><code> )</code>
	88	<p>
	89	A <strong>Steiner system</strong> is a <var>t</var>-(<var>v</var>,<var>k</var>,1) design, and in this case it
	90	is possible to extend the notion of intersection triangle defined in
	91	<a href="CHAP002.htm#SSEC001.3">TDesignIntersectionTriangle</a>.
	92	<p>
	93	Suppose <i>D</i> is a <var>t</var>-(<var>v</var>,<var>k</var>,1) design, with <i>B</i> a block of <i>D</i>,
	94	let <i>i</i> and <i>j</i> be non-negative integers with <i>i</i>+<i>j</i> ≤ <i>k</i>, and suppose
	95	<i>X</i> and <i>Y</i> are disjoint subsets of <i>B</i>, such that <i>X</i> and <i>Y</i> have
	96	respective sizes <i>i</i> and <i>j</i>. The (<i>i</i>,<i>j</i>)-<strong>intersection number</strong> is the
	97	number of blocks of <i>D</i> that contain <i>X</i> and are disjoint from <i>Y</i>
	98	(this number depends only on <var>t</var>, <var>v</var>, <var>k</var>, <i>i</i> and <i>j</i>). Note that
	99	when <i>i</i>+<i>j</i> ≤ <i>t</i>, this intersection number is the same as that defined in
	100	<a href="CHAP002.htm#SSEC001.3">TDesignIntersectionTriangle</a> for the general <var>t</var>-design case.
	101	<p>
	102	Given a non-negative integer <var>t</var>, and positive integers <var>v</var> and
	103	<var>k</var>, with <i>t</i> ≤ <i>k</i> ≤ <i>v</i> , this function returns the <strong>Steiner
	104	system intersection triangle</strong>, which is a two dimensional array whose
	105	(<i>i</i>+1,<i>j</i>+1)-entry is the (<i>i</i>,<i>j</i>)-intersection number for a <var>t</var>-(<var>v</var>,<var>k</var>,1)
	106	design (assuming such a design exists), such that <i>i</i>,<i>j</i> ≥ 0, <i>i</i>+<i>j</i> ≤ <i>k</i>. This function returns <code>fail</code> if <code>TDesignLambdas(</code><var>t</var><code>,</code><var>v</var><code>,</code><var>k</var><code>,1)</code> does.
	107	<p>
	108	See also <a href="CHAP002.htm#SSEC001.3">TDesignIntersectionTriangle</a>.
	109	<p>
	110	<pre>
	111	gap> SteinerSystemIntersectionTriangle(5,24,8);
	112	[ [ 759, 506, 330, 210, 130, 78, 46, 30, 30 ],
	113	[ 253, 176, 120, 80, 52, 32, 16, 0 ], [ 77, 56, 40, 28, 20, 16, 16 ],
	114	[ 21, 16, 12, 8, 4, 0 ], [ 5, 4, 4, 4, 4 ], [ 1, 0, 0, 0 ], [ 1, 0, 0 ],
	115	[ 1, 0 ], [ 1 ] ]
	116	gap> TDesignIntersectionTriangle(5,24,8,1);
	117	[ [ 759, 506, 330, 210, 130, 78 ], [ 253, 176, 120, 80, 52 ],
	118	[ 77, 56, 40, 28 ], [ 21, 16, 12 ], [ 5, 4 ], [ 1 ] ]
	119	</pre>
	120	<p>
	121	<a name = "SSEC001.5"></a>
	122	<li><code>TDesignBlockMultiplicityBound( </code><var>t</var><code>, </code><var>v</var><code>, </code><var>k</var><code>, </code><var>lambda</var><code> )</code>
	123	<p>
	124	Given a non-negative integer <var>t</var>, and positive integers <var>v</var>, <var>k</var> and
	125	<var>lambda</var>, with <i>t</i> ≤ <i>k</i> ≤ <i>v</i> , this function returns a non-negative
	126	integer which is an upper bound on the multiplicity of any block in
	127	any <var>t</var>-(<var>v</var>,<var>k</var>,<var>lambda</var>) design (the <strong>multiplicity</strong> of a block in
	128	a block design is the number of times that block occurs in the block
	129	list). In particular, if the value 0 is returned, then this implies
	130	that a <var>t</var>-(<var>v</var>,<var>k</var>,<var>lambda</var>) design does not exist.
	131	<p>
	132	Although our bounds are reasonably good, we do not claim that the
	133	returned bound <i>m</i> is always achieved; that is, there may not exist a
	134	<var>t</var>-(<var>v</var>,<var>k</var>,<var>lambda</var>) design having a block with multiplicity <i>m</i>.
	135	<p>
	136	See also <a href="CHAP002.htm#SSEC001.6">ResolvableTDesignBlockMultiplicityBound</a>.
	137	<p>
	138	<pre>
	139	gap> TDesignBlockMultiplicityBound(5,16,7,5);
	140	2
	141	gap> TDesignBlockMultiplicityBound(2,36,6,1);
	142	0
	143	gap> TDesignBlockMultiplicityBound(2,36,6,2);
	144	2
	145	gap> TDesignBlockMultiplicityBound(2,15,5,2);
	146	0
	147	gap> TDesignBlockMultiplicityBound(2,15,5,4);
	148	2
	149	gap> TDesignBlockMultiplicityBound(2,11,4,6);
	150	3
	151	</pre>
	152	<p>
	153	<a name = "SSEC001.6"></a>
	154	<li><code>ResolvableTDesignBlockMultiplicityBound( </code><var>t</var><code>, </code><var>v</var><code>, </code><var>k</var><code>, </code><var>lambda</var><code> )</code>
	155	<p>
	156	A <strong>resolution</strong> of a block design is a partition of the blocks into
	157	subsets, each of which forms a partition of the point set, and a block
	158	design is <strong>resolvable</strong> if it has a resolution.
	159	<p>
	160	Given a non-negative integer <var>t</var>, and positive integers <var>v</var>, <var>k</var> and
	161	<var>lambda</var>, with <i>t</i> ≤ <i>k</i> ≤ <i>v</i> , this function returns a non-negative
	162	integer which is an upper bound on the multiplicity of any block in any
	163	resolvable <var>t</var>-(<var>v</var>,<var>k</var>,<var>lambda</var>) design (the <strong>multiplicity</strong> of a block
	164	in a block design is the number of times that block occurs in the block
	165	list). In particular, if the value 0 is returned, then this implies
	166	that a resolvable <var>t</var>-(<var>v</var>,<var>k</var>,<var>lambda</var>) design does not exist.
	167	<p>
	168	Although our bounds are reasonably good, we do not claim that the returned
	169	bound <i>m</i> is always achieved; that is, there may not exist a resolvable
	170	<var>t</var>-(<var>v</var>,<var>k</var>,<var>lambda</var>) design having a block with multiplicity <i>m</i>.
	171	<p>
	172	See also <a href="CHAP002.htm#SSEC001.5">TDesignBlockMultiplicityBound</a>.
	173	<p>
	174	<pre>
	175	gap> ResolvableTDesignBlockMultiplicityBound(5,12,6,1);
	176	1
	177	gap> ResolvableTDesignBlockMultiplicityBound(2,21,7,3);
	178	0
	179	gap> TDesignBlockMultiplicityBound(2,21,7,3);
	180	1
	181	gap> ResolvableTDesignBlockMultiplicityBound(2,12,4,3);
	182	1
	183	gap> TDesignBlockMultiplicityBound(2,12,4,3);
	184	2
	185	</pre>
	186	<p>
	187	<p>
	188	<h2><a name="SECT002">2.2 Block intersection polynomials</a></h2>
	189	<p><p>
	190	In <a href="biblio.htm#CaSo"><cite>CaSo</cite></a>, Cameron and Soicher introduce block intersection
	191	polynomials and their applications to the study of block designs.
	192	Here we give functions to construct and analyze block intersection
	193	polynomials.
	194	<p>
	195	<a name = "SSEC002.1"></a>
	196	<li><code>BlockIntersectionPolynomial(</code><var>x</var><code>, </code><var>m</var><code>, </code><var>lambdavec</var><code> )</code>
	197	<p>
	198	For <i>k</i> a non-negative integer, define the polynomial
	199	<i>P</i>(<i>x</i>,<i>k</i>)=<i>x</i>(<i>x</i>−1)…(<i>x</i>−<i>k</i>+1). Let <i>s</i> and <i>t</i> be non-negative
	200	integers, with <i>s</i> ≥ <i>t</i>, and let <i>m</i><sub>0</sub>,…,<i>m</i><sub><i>s</i></sub> and
	201	λ<sub>0</sub>,…,λ<sub><i>t</i></sub> be rational numbers. Then the <strong>block
	202	intersection polynomial</strong> for the sequences [<i>m</i><sub>0</sub>,…,<i>m</i><sub><i>s</i></sub>],
	203	[λ<sub>0</sub>,…,λ<sub><i>t</i></sub>] is defined to be
	204	<br clear="all" /><table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"></td><td nowrap="nowrap" align="center"><small><i>t</i></small><!--sup--><br /><span class="largerstill">∑<br /></span><small><i>j</i>=0</small> <br /></td><td nowrap="nowrap" align="center"></td><td align="left" class="cl">⎛<br />⎝</td><td nowrap="nowrap" align="center"><i>t</i><br /><i>j</i><br /></td><td align="left" class="cl">⎞<br />⎠</td><td nowrap="nowrap" align="center"><i>P</i>(−<i>x</i>,<i>t</i>−<i>j</i>)[<i>P</i>(<i>s</i>,<i>j</i>)λ<sub><i>j</i></sub>−</td><td nowrap="nowrap" align="center"><small><i>s</i></small><!--sup--><br /><span class="largerstill">∑<br /></span><small><i>i</i>=<i>j</i></small> <br /></td><td nowrap="nowrap" align="center"><i>P</i>(<i>i</i>,<i>j</i>)<i>m</i><sub><i>i</i></sub>],</td></tr></table></td></tr></table>
	205	and is denoted by <i>B</i>(<i>x</i>,[<i>m</i><sub>0</sub>,…,<i>m</i><sub><i>s</i></sub>],[λ<sub>0</sub>,…,λ<sub><i>t</i></sub>])·
	206	<p>
	207	Now suppose <var>x</var> is an indeterminate over the rationals, and <var>m</var> and
	208	<var>lambdavec</var> are non-empty lists of rational numbers, such that the length
	209	of <var>lambdavec</var> is not greater than that of <var>m</var>. Then this function
	210	returns the block intersection polynomial <i>B</i>(<i>x</i> ,<i>m</i> ,<i>lambdavec</i> ).
	211	<p>
	212	The importance of a block intersection polynomial is as follows.
	213	Let <i>D</i>=(<i>V</i>,<span style="font-family:helvetica"><i>B</i></span>) be a block design, let <i>S</i> ⊆ <i>V</i>, with <i>s</i>=\|<i>S</i>\|,
	214	and for <i>i</i>=0,…,<i>s</i>, suppose that <i>m</i><sub><i>i</i></sub> is a non-negative integer
	215	with <i>m</i><sub><i>i</i></sub> ≤ <i>n</i><sub><i>i</i></sub>, where <i>n</i><sub><i>i</i></sub> is the number of blocks intersecting <i>S</i>
	216	in exactly <i>i</i> points. Let <i>t</i> be a non-negative <strong>even</strong> integer with <i>t</i> ≤ <i>s</i>, and suppose that, for <i>j</i>=0…,<i>t</i>, we have <br clear="all" /><table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center">λ<sub><i>j</i></sub>=1/</td><td align="left" class="cl">⎛<br />⎝</td><td nowrap="nowrap" align="center"><i>s</i><br /><i>j</i><br /></td><td align="left" class="cl">⎞<br />⎠</td><td nowrap="nowrap" align="center"></td><td nowrap="nowrap" align="center"><small></small><!--sup--><br /><span class="largerstill">∑<br /></span><small><i>T</i> ⊆ <i>S</i>,\|<i>T</i>\|=<i>j</i></small> <br /></td><td nowrap="nowrap" align="center">λ<sub><i>T</i></sub></td></tr></table></td></tr></table>, where λ<sub><i>T</i></sub> is the
	217	number of blocks of <i>D</i> containing <i>T</i>. Then the block intersection
	218	polynomial <i>B</i>(<i>x</i>)=<i>B</i>(<i>x</i>,[<i>m</i><sub>0</sub>,…,<i>m</i><sub><i>s</i></sub>],[λ<sub>0</sub>,…,λ<sub><i>t</i></sub>])
	219	is a polynomial with integer coefficients, and <i>B</i>(<i>n</i>) ≥ 0 for every
	220	integer <i>n</i>. (These conditions can be checked using the function
	221	<a href="CHAP002.htm#SSEC002.2">BlockIntersectionPolynomialCheck</a>.) In addition, if <i>B</i>(<i>n</i>)=0 for some
	222	integer <i>n</i>, then <i>m</i><sub><i>i</i></sub>=<i>n</i><sub><i>i</i></sub> for <i>i</i> ∉ {<i>n</i>,<i>n</i>+1,…,<i>n</i>+<i>t</i>−1}.
	223	<p>
	224	For more information on block intersection polynomials and their
	225	applications, see <a href="biblio.htm#CaSo"><cite>CaSo</cite></a> and <a href="biblio.htm#Soi1"><cite>Soi1</cite></a>.
	226	<p>
	227	<pre>
	228	gap> x:=Indeterminate(Rationals,1);
	229	x_1
	230	gap> m:=[0,0,0,0,0,0,0,1];;
	231	gap> lambdavec:=TDesignLambdas(6,14,7,4);
	232	[ 1716, 858, 396, 165, 60, 18, 4 ]
	233	gap> B:=BlockIntersectionPolynomial(x,m,lambdavec);
	234	1715x_1^6-10269x_1^5+34685x_1^4-69615x_1^3+84560x_1^2-56196x_1+15120
	235	gap> Factors(B);
	236	[ 1715*x_1-1715,
	237	x_1^5-1222/245x_1^4+3733/245x_1^3-6212/245x_1^2+5868/245x_1-432/49 ]
	238	gap> Value(B,1);
	239	0
	240	</pre>
	241	<p>
	242	<a name = "SSEC002.2"></a>
	243	<li><code>BlockIntersectionPolynomialCheck(</code><var>m</var><code>, </code><var>lambdavec</var><code>)</code>
	244	<p>
	245	Suppose <var>m</var> is a list of non-negative integers, and <var>lambdavec</var> is a
	246	list of non-negative rational numbers, with the length of <var>lambdavec</var>
	247	odd and not greater than the length of <var>m</var>.
	248	<p>
	249	Then, for <var>x</var> an indeterminate over the rationals, this function
	250	checks whether <code>BlockIntersectionPolynomial(</code><var>x</var><code>,</code><var>m</var><code>,</code><var>lambdavec</var><code>)</code> is a
	251	polynomial over the integers and has a non-negative value at each integer.
	252	The function returns <code>true</code> if this is so; else <code>false</code> is returned.
	253	<p>
	254	See also <a href="CHAP002.htm#SSEC002.1">BlockIntersectionPolynomial</a>.
	255	<p>
	256	<pre>
	257	gap> m:=[0,0,0,0,0,0,0,1];;
	258	gap> lambdavec:=TDesignLambdas(6,14,7,4);
	259	[ 1716, 858, 396, 165, 60, 18, 4 ]
	260	gap> BlockIntersectionPolynomialCheck(m,lambdavec);
	261	true
	262	gap> m:=[1,0,0,0,0,0,0,1];;
	263	gap> BlockIntersectionPolynomialCheck(m,lambdavec);
	264	false
	265	</pre>
	266	<p>
	267	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP001.htm">Previous</a>] [<a href ="CHAP003.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
	268	<P>
	269	<address>design manual<br>March 2019
	270	</address></body></html>⏎

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	0	<html><head><title>[design] 3 Constructing block designs</title></head>
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	3	<h1>3 Constructing block designs</h1><p>
	4	<P>
	5	<H3>Sections</H3>
	6	<oL>
	7	<li> <A HREF="CHAP003.htm#SECT001">Functions to construct block designs</a>
	8	</ol><p>
	9	<p>
	10	<p>
	11	<h2><a name="SECT001">3.1 Functions to construct block designs</a></h2>
	12	<p><p>
	13	<a name = "SSEC001.1"></a>
	14	<li><code>BlockDesign( </code><var>v</var><code>, </code><var>B</var><code> )</code>
	15	<li><code>BlockDesign( </code><var>v</var><code>, </code><var>B</var><code>, </code><var>G</var><code> )</code>
	16	<p>
	17	Let <var>v</var> be a positive integer and <var>B</var> a non-empty list of non-empty
	18	sorted lists of elements of {1,…,<i>v</i> }.
	19	<p>
	20	The first version of this function returns the block design with point-set
	21	{1,…,<i>v</i> } and block multiset <var>C</var>, where <var>C</var> is <code>SortedList(</code><var>B</var><code>)</code>.
	22	<p>
	23	For the second version of this function, we require <var>G</var> to be a group
	24	of permutations of {1,…,<i>v</i> }, and the function returns the
	25	block design with point-set {1,…,<i>v</i> } and block multiset <var>C</var>,
	26	where <var>C</var> is the sorted list of the concatenation of the <var>G</var>-orbits of
	27	the elements of <var>B</var>.
	28	<p>
	29	<pre>
	30	gap> BlockDesign( 2, [[1,2],[1],[1,2]] );
	31	rec( isBlockDesign := true, v := 2, blocks := [ [ 1 ], [ 1, 2 ], [ 1, 2 ] ] )
	32	gap> D:=BlockDesign(7, [[1,2,4]], Group((1,2,3,4,5,6,7)));
	33	rec( isBlockDesign := true, v := 7,
	34	blocks := [ [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 5, 6 ], [ 2, 3, 5 ],
	35	[ 2, 6, 7 ], [ 3, 4, 6 ], [ 4, 5, 7 ] ],
	36	autSubgroup := Group([ (1,2,3,4,5,6,7) ]) )
	37	gap> AllTDesignLambdas(D);
	38	[ 7, 3, 1 ]
	39	</pre>
	40	<p>
	41	<a name = "SSEC001.2"></a>
	42	<li><code>AGPointFlatBlockDesign( </code><var>n</var><code>, </code><var>q</var><code>, </code><var>d</var><code> )</code>
	43	<p>
	44	Let <var>n</var> be positive integer, <var>q</var> a prime-power, and <var>d</var> a non-negative
	45	integer less than or equal to <var>n</var>. Then this function returns the block
	46	design whose points are the points of the affine space <i>AG</i>(<i>n</i> ,<i>q</i> ),
	47	and whose blocks are the <i>d</i> -flats of <i>AG</i>(<i>n</i> ,<i>q</i> ), considering a
	48	<i>d</i> -flat as a set of points.
	49	<p>
	50	Note that the <strong>affine space</strong> <i>AG</i>(<i>n</i> ,<i>q</i> ) consists of all the cosets of
	51	all the subspaces of the vector space <i>V</i>(<i>n</i> ,<i>q</i> ), with the <strong>points</strong>
	52	being the cosets of the 0-dimensional subspace and the <strong><i>d</i> -flats</strong>
	53	being the cosets of the <i>d</i> -dimensional subspaces. As is usual, we
	54	identify the points with the vectors in <i>V</i>(<i>n</i> ,<i>q</i> ), and these vectors
	55	are given as the point-names.
	56	<p>
	57	<pre>
	58	gap> D:=AGPointFlatBlockDesign(2,4,1);
	59	rec( isBlockDesign := true, v := 16,
	60	blocks := [ [ 1, 2, 3, 4 ], [ 1, 5, 9, 13 ], [ 1, 6, 11, 16 ],
	61	[ 1, 7, 12, 14 ], [ 1, 8, 10, 15 ], [ 2, 5, 12, 15 ], [ 2, 6, 10, 14 ],
	62	[ 2, 7, 9, 16 ], [ 2, 8, 11, 13 ], [ 3, 5, 10, 16 ], [ 3, 6, 12, 13 ],
	63	[ 3, 7, 11, 15 ], [ 3, 8, 9, 14 ], [ 4, 5, 11, 14 ], [ 4, 6, 9, 15 ],
	64	[ 4, 7, 10, 13 ], [ 4, 8, 12, 16 ], [ 5, 6, 7, 8 ], [ 9, 10, 11, 12 ],
	65	[ 13, 14, 15, 16 ] ],
	66	autSubgroup := Group([ (5,9,13)(6,10,14)(7,11,15)(8,12,16),
	67	(2,5,6)(3,9,11)(4,13,16)(7,14,12)(8,10,15),
	68	(1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16),
	69	(3,4)(7,8)(9,13)(10,14)(11,16)(12,15) ]),
	70	pointNames := [ [ 0Z(2), 0Z(2) ], [ 0Z(2), Z(2)^0 ], [ 0Z(2), Z(2^2) ],
	71	[ 0Z(2), Z(2^2)^2 ], [ Z(2)^0, 0Z(2) ], [ Z(2)^0, Z(2)^0 ],
	72	[ Z(2)^0, Z(2^2) ], [ Z(2)^0, Z(2^2)^2 ], [ Z(2^2), 0*Z(2) ],
	73	[ Z(2^2), Z(2)^0 ], [ Z(2^2), Z(2^2) ], [ Z(2^2), Z(2^2)^2 ],
	74	[ Z(2^2)^2, 0*Z(2) ], [ Z(2^2)^2, Z(2)^0 ], [ Z(2^2)^2, Z(2^2) ],
	75	[ Z(2^2)^2, Z(2^2)^2 ] ] )
	76	gap> AllTDesignLambdas(D);
	77	[ 20, 5, 1 ]
	78	</pre>
	79	<p>
	80	<a name = "SSEC001.3"></a>
	81	<li><code>PGPointFlatBlockDesign( </code><var>n</var><code>, </code><var>q</var><code>, </code><var>d</var><code> )</code>
	82	<p>
	83	Let <var>n</var> be a non-negative integer, <var>q</var> a prime-power, and <var>d</var> a
	84	non-negative integer less than or equal to <var>n</var>. Then this function
	85	returns the block design whose points are the (projective) points of
	86	the projective space <i>PG</i>(<i>n</i> ,<i>q</i> ), and whose blocks are the <i>d</i> -flats
	87	of <i>PG</i>(<i>n</i> ,<i>q</i> ), considering a <i>d</i> -flat as a set of projective points.
	88	<p>
	89	Note that the <strong>projective space</strong> <i>PG</i>(<i>n</i> ,<i>q</i> ) consists of all the
	90	subspaces of the vector space <i>V</i>(<i>n</i> +1,<i>q</i> ), with the <strong>projective
	91	points</strong> being the 1-dimensional subspaces and the <strong><i>d</i> -flats</strong> being
	92	the (<i>d</i> +1)-dimensional subspaces.
	93	<p>
	94	<pre>
	95	gap> D:=PGPointFlatBlockDesign(3,2,1);
	96	rec( isBlockDesign := true, v := 15,
	97	blocks := [ [ 1, 2, 3 ], [ 1, 4, 5 ], [ 1, 6, 7 ], [ 1, 8, 9 ],
	98	[ 1, 10, 11 ], [ 1, 12, 13 ], [ 1, 14, 15 ], [ 2, 4, 6 ], [ 2, 5, 7 ],
	99	[ 2, 8, 10 ], [ 2, 9, 11 ], [ 2, 12, 14 ], [ 2, 13, 15 ], [ 3, 4, 7 ],
	100	[ 3, 5, 6 ], [ 3, 8, 11 ], [ 3, 9, 10 ], [ 3, 12, 15 ], [ 3, 13, 14 ],
	101	[ 4, 8, 12 ], [ 4, 9, 13 ], [ 4, 10, 14 ], [ 4, 11, 15 ], [ 5, 8, 13 ],
	102	[ 5, 9, 12 ], [ 5, 10, 15 ], [ 5, 11, 14 ], [ 6, 8, 14 ], [ 6, 9, 15 ],
	103	[ 6, 10, 12 ], [ 6, 11, 13 ], [ 7, 8, 15 ], [ 7, 9, 14 ],
	104	[ 7, 10, 13 ], [ 7, 11, 12 ] ],
	105	autSubgroup := Group([ (8,12)(9,13)(10,14)(11,15),
	106	(1,2,4,8)(3,6,12,9)(5,10)(7,14,13,11) ]),
	107	pointNames := [ <vector space of dimension 1 over GF(2)>,
	108	<vector space of dimension 1 over GF(2)>,
	109	<vector space of dimension 1 over GF(2)>,
	110	<vector space of dimension 1 over GF(2)>,
	111	<vector space of dimension 1 over GF(2)>,
	112	<vector space of dimension 1 over GF(2)>,
	113	<vector space of dimension 1 over GF(2)>,
	114	<vector space of dimension 1 over GF(2)>,
	115	<vector space of dimension 1 over GF(2)>,
	116	<vector space of dimension 1 over GF(2)>,
	117	<vector space of dimension 1 over GF(2)>,
	118	<vector space of dimension 1 over GF(2)>,
	119	<vector space of dimension 1 over GF(2)>,
	120	<vector space of dimension 1 over GF(2)>,
	121	<vector space of dimension 1 over GF(2)> ] )
	122	gap> AllTDesignLambdas(D);
	123	[ 35, 7, 1 ]
	124	</pre>
	125	<p>
	126	<a name = "SSEC001.4"></a>
	127	<li><code>WittDesign( </code><var>n</var><code> )</code>
	128	<p>
	129	Suppose <i>n</i> ∈ {9,10,11,12,21,22,23,24}.
	130	<p>
	131	If <i>n</i> =24 then this function returns the <strong>large Witt design</strong> <i>W</i><sub>24</sub>,
	132	the unique (up to isomorphism) 5-(24,8,1) design. If <i>n</i> =24−<i>i</i>, where
	133	<i>i</i> ∈ {1,2,3}, then the <i>i</i>-fold point-derived design of <i>W</i><sub>24</sub>
	134	is returned; this is the unique (up to isomorphism) (5−<i>i</i>)-(24−<i>i</i>,8−<i>i</i>,1)
	135	design.
	136	<p>
	137	If <i>n</i> =12 then this function returns the <strong>small Witt design</strong> <i>W</i><sub>12</sub>,
	138	the unique (up to isomorphism) 5-(12,6,1) design. If <i>n</i> =12−<i>i</i>, where
	139	<i>i</i> ∈ {1,2,3}, then the <i>i</i>-fold point-derived design of <i>W</i><sub>12</sub>
	140	is returned; this is the unique (up to isomorphism) (5−<i>i</i>)-(12−<i>i</i>,6−<i>i</i>,1)
	141	design.
	142	<p>
	143	<pre>
	144	gap> W24:=WittDesign(24);;
	145	gap> AllTDesignLambdas(W24);
	146	[ 759, 253, 77, 21, 5, 1 ]
	147	gap> DisplayCompositionSeries(AutomorphismGroup(W24));
	148	G (3 gens, size 244823040)
	149	\| M(24)
	150	1 (0 gens, size 1)
	151	gap> W10:=WittDesign(10);;
	152	gap> AllTDesignLambdas(W10);
	153	[ 30, 12, 4, 1 ]
	154	gap> DisplayCompositionSeries(AutomorphismGroup(W10));
	155	G (4 gens, size 1440)
	156	\| Z(2)
	157	S (4 gens, size 720)
	158	\| Z(2)
	159	S (3 gens, size 360)
	160	\| A(6) ~ A(1,9) = L(2,9) ~ B(1,9) = O(3,9) ~ C(1,9) = S(2,9) ~ 2A(1,9) = U(2,\
	161	9)
	162	1 (0 gens, size 1)
	163	</pre>
	164	<p>
	165	<a name = "SSEC001.5"></a>
	166	<li><code>DualBlockDesign( </code><var>D</var><code> )</code>
	167	<p>
	168	Suppose <var>D</var> is a block design for which every point lies on at least
	169	one block. Then this function returns the dual of <var>D</var>, the block design
	170	in which the roles of points and blocks are interchanged, but incidence
	171	(including repeated incidence) stays the same. Note that, since the
	172	list of blocks of a block design is always sorted, the block list of
	173	the dual of the dual of <var>D</var> may not be equal to the block list of <var>D</var>.
	174	<p>
	175	<pre>
	176	gap> D:=BlockDesign(4,[[1,3],[2,3,4],[3,4]]);;
	177	gap> dualD:=DualBlockDesign(D);
	178	rec( isBlockDesign := true, v := 3,
	179	blocks := [ [ 1 ], [ 1, 2, 3 ], [ 2 ], [ 2, 3 ] ],
	180	pointNames := [ [ 1, 3 ], [ 2, 3, 4 ], [ 3, 4 ] ] )
	181	gap> DualBlockDesign(dualD).blocks;
	182	[ [ 1, 2 ], [ 2, 3, 4 ], [ 2, 4 ] ]
	183	</pre>
	184	<p>
	185	<a name = "SSEC001.6"></a>
	186	<li><code>ComplementBlocksBlockDesign( </code><var>D</var><code> )</code>
	187	<p>
	188	Suppose <var>D</var> is a binary incomplete-block design.
	189	Then this function returns the block design on the same
	190	point-set as <var>D</var>, whose blocks are the complements of
	191	those of <var>D</var> (complemented with respect to the point-set).
	192	<p>
	193	<pre>
	194	gap> D:=PGPointFlatBlockDesign(2,2,1);
	195	rec( isBlockDesign := true, v := 7,
	196	pointNames := [ <vector space of dimension 1 over GF(2)>,
	197	<vector space of dimension 1 over GF(2)>,
	198	<vector space of dimension 1 over GF(2)>,
	199	<vector space of dimension 1 over GF(2)>,
	200	<vector space of dimension 1 over GF(2)>,
	201	<vector space of dimension 1 over GF(2)>,
	202	<vector space of dimension 1 over GF(2)> ],
	203	blocks := [ [ 1, 2, 3 ], [ 1, 4, 5 ], [ 1, 6, 7 ], [ 2, 4, 6 ],
	204	[ 2, 5, 7 ], [ 3, 4, 7 ], [ 3, 5, 6 ] ] )
	205	gap> AllTDesignLambdas(D);
	206	[ 7, 3, 1 ]
	207	gap> C:=ComplementBlocksBlockDesign(D);
	208	rec( isBlockDesign := true, v := 7,
	209	blocks := [ [ 1, 2, 4, 7 ], [ 1, 2, 5, 6 ], [ 1, 3, 4, 6 ], [ 1, 3, 5, 7 ],
	210	[ 2, 3, 4, 5 ], [ 2, 3, 6, 7 ], [ 4, 5, 6, 7 ] ],
	211	pointNames := [ <vector space of dimension 1 over GF(2)>,
	212	<vector space of dimension 1 over GF(2)>,
	213	<vector space of dimension 1 over GF(2)>,
	214	<vector space of dimension 1 over GF(2)>,
	215	<vector space of dimension 1 over GF(2)>,
	216	<vector space of dimension 1 over GF(2)>,
	217	<vector space of dimension 1 over GF(2)> ] )
	218	gap> AllTDesignLambdas(C);
	219	[ 7, 4, 2 ]
	220	</pre>
	221	<p>
	222	<a name = "SSEC001.7"></a>
	223	<li><code>DeletedPointsBlockDesign( </code><var>D</var><code>, </code><var>Y</var><code> )</code>
	224	<p>
	225	Suppose <var>D</var> is a block design and <var>Y</var> is a proper subset of the point-set
	226	of <var>D</var>.
	227	<p>
	228	Then this function returns the block design <i>DP</i> obtained from <var>D</var> by
	229	deleting the points in <var>Y</var> from the point-set, and from each block.
	230	It is an error if the resulting design contains an empty block.
	231	The points of <i>DP</i> are relabelled 1,2,·.·, preserving the order of
	232	the corresponding points of <var>D</var>; the point-names of <i>DP</i> (listed in
	233	<code><i>DP</i>.pointNames</code>) are those of these corresponding points of <var>D</var>.
	234	<p>
	235	<pre>
	236	gap> D:=BlockDesigns(rec(v:=11,blockSizes:=[5],
	237	> tSubsetStructure:=rec(t:=2,lambdas:=[2])))[1];
	238	rec( isBlockDesign := true, v := 11,
	239	blocks := [ [ 1, 2, 3, 4, 5 ], [ 1, 2, 9, 10, 11 ], [ 1, 3, 6, 7, 9 ],
	240	[ 1, 4, 7, 8, 10 ], [ 1, 5, 6, 8, 11 ], [ 2, 3, 6, 8, 10 ],
	241	[ 2, 4, 6, 7, 11 ], [ 2, 5, 7, 8, 9 ], [ 3, 4, 8, 9, 11 ],
	242	[ 3, 5, 7, 10, 11 ], [ 4, 5, 6, 9, 10 ] ],
	243	tSubsetStructure := rec( t := 2, lambdas := [ 2 ] ), isBinary := true,
	244	isSimple := true, blockSizes := [ 5 ], blockNumbers := [ 11 ], r := 5,
	245	autGroup := Group([ (2,4)(3,5)(7,11)(8,9), (1,3)(2,5)(7,9)(10,11),
	246	(1,5,3)(6,11,7)(8,10,9), (1,10,5,2,11,3)(4,9,7)(6,8) ]) )
	247	gap> AllTDesignLambdas(D);
	248	[ 11, 5, 2 ]
	249	gap> DP:=DeletedPointsBlockDesign(D,[5,8]);
	250	rec( isBlockDesign := true, v := 9,
	251	blocks := [ [ 1, 2, 3, 4 ], [ 1, 2, 7, 8, 9 ], [ 1, 3, 5, 6, 7 ],
	252	[ 1, 4, 6, 8 ], [ 1, 5, 9 ], [ 2, 3, 5, 8 ], [ 2, 4, 5, 6, 9 ],
	253	[ 2, 6, 7 ], [ 3, 4, 7, 9 ], [ 3, 6, 8, 9 ], [ 4, 5, 7, 8 ] ],
	254	pointNames := [ 1, 2, 3, 4, 6, 7, 9, 10, 11 ] )
	255	gap> PairwiseBalancedLambda(DP);
	256	2
	257	</pre>
	258	<p>
	259	<a name = "SSEC001.8"></a>
	260	<li><code>DeletedBlocksBlockDesign( </code><var>D</var><code>, </code><var>Y</var><code>)</code>
	261	<p>
	262	Suppose <var>D</var> is a block design, and <var>Y</var> is a proper sublist of the
	263	block-list of <var>D</var> (<var>Y</var> need not be sorted).
	264	<p>
	265	Then this function returns the block design obtained from <var>D</var> by deleting
	266	the blocks in <var>Y</var> (counting repeats) from the block-list of <var>D</var>.
	267	<p>
	268	<pre>
	269	gap> D:=BlockDesign(7,[[1,2,4],[1,2,4]],Group((1,2,3,4,5,6,7)));
	270	rec( isBlockDesign := true, v := 7,
	271	blocks := [ [ 1, 2, 4 ], [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 3, 7 ],
	272	[ 1, 5, 6 ], [ 1, 5, 6 ], [ 2, 3, 5 ], [ 2, 3, 5 ], [ 2, 6, 7 ],
	273	[ 2, 6, 7 ], [ 3, 4, 6 ], [ 3, 4, 6 ], [ 4, 5, 7 ], [ 4, 5, 7 ] ],
	274	autSubgroup := Group([ (1,2,3,4,5,6,7) ]) )
	275	gap> DeletedBlocksBlockDesign(D,[[2,3,5],[2,3,5],[4,5,7]]);
	276	rec( isBlockDesign := true, v := 7,
	277	blocks := [ [ 1, 2, 4 ], [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 3, 7 ],
	278	[ 1, 5, 6 ], [ 1, 5, 6 ], [ 2, 6, 7 ], [ 2, 6, 7 ], [ 3, 4, 6 ],
	279	[ 3, 4, 6 ], [ 4, 5, 7 ] ] )
	280	</pre>
	281	<p>
	282	<a name = "SSEC001.9"></a>
	283	<li><code>AddedPointBlockDesign( </code><var>D</var><code>, </code><var>Y</var><code> )</code>
	284	<li><code>AddedPointBlockDesign( </code><var>D</var><code>, </code><var>Y</var><code>, </code><var>pointname</var><code> )</code>
	285	<p>
	286	Suppose <var>D</var> is a block design, and <var>Y</var> is a sublist of the block-list
	287	of <var>D</var> (<var>Y</var> need not be sorted).
	288	<p>
	289	Then this function returns the block design obtained from <var>D</var> by adding
	290	the new point <code></code><var>D</var><code>.v+1</code> to the point-set, and adding this new point
	291	(once) to each block of <var>Y</var> (where repeats count).
	292	<p>
	293	The optional parameter <var>pointname</var> specifies a point-name for the
	294	new point.
	295	<p>
	296	<pre>
	297	gap> D:=BlockDesign(7,[[1,2,4],[1,2,4]],Group((1,2,3,4,5,6,7)));
	298	rec( isBlockDesign := true, v := 7,
	299	blocks := [ [ 1, 2, 4 ], [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 3, 7 ],
	300	[ 1, 5, 6 ], [ 1, 5, 6 ], [ 2, 3, 5 ], [ 2, 3, 5 ], [ 2, 6, 7 ],
	301	[ 2, 6, 7 ], [ 3, 4, 6 ], [ 3, 4, 6 ], [ 4, 5, 7 ], [ 4, 5, 7 ] ],
	302	autSubgroup := Group([ (1,2,3,4,5,6,7) ]) )
	303	gap> AddedPointBlockDesign(D,[[2,3,5],[2,3,5],[4,5,7]],"infinity");
	304	rec( isBlockDesign := true, v := 8,
	305	blocks := [ [ 1, 2, 4 ], [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 3, 7 ],
	306	[ 1, 5, 6 ], [ 1, 5, 6 ], [ 2, 3, 5, 8 ], [ 2, 3, 5, 8 ], [ 2, 6, 7 ],
	307	[ 2, 6, 7 ], [ 3, 4, 6 ], [ 3, 4, 6 ], [ 4, 5, 7 ], [ 4, 5, 7, 8 ] ],
	308	pointNames := [ 1, 2, 3, 4, 5, 6, 7, "infinity" ] )
	309	</pre>
	310	<p>
	311	<a name = "SSEC001.10"></a>
	312	<li><code>AddedBlocksBlockDesign( </code><var>D</var><code>, </code><var>Y</var><code> )</code>
	313	<p>
	314	Suppose <var>Y</var> is a list of multisets of points of the block design <var>D</var>.
	315	Then this function returns a new block design, whose point-set is that
	316	of <var>D</var>, and whose block list is that of <var>D</var> with the elements of <var>Y</var>
	317	(including repeats) added.
	318	<p>
	319	<pre>
	320	gap> D:=BlockDesign(7,[[1,2,4]],Group((1,2,3,4,5,6,7)));
	321	rec( isBlockDesign := true, v := 7,
	322	blocks := [ [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 5, 6 ], [ 2, 3, 5 ],
	323	[ 2, 6, 7 ], [ 3, 4, 6 ], [ 4, 5, 7 ] ],
	324	autSubgroup := Group([ (1,2,3,4,5,6,7) ]) )
	325	gap> AddedBlocksBlockDesign(D,D.blocks);
	326	rec( isBlockDesign := true, v := 7,
	327	blocks := [ [ 1, 2, 4 ], [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 3, 7 ],
	328	[ 1, 5, 6 ], [ 1, 5, 6 ], [ 2, 3, 5 ], [ 2, 3, 5 ], [ 2, 6, 7 ],
	329	[ 2, 6, 7 ], [ 3, 4, 6 ], [ 3, 4, 6 ], [ 4, 5, 7 ], [ 4, 5, 7 ] ] )
	330	</pre>
	331	<p>
	332	<a name = "SSEC001.11"></a>
	333	<li><code>DerivedBlockDesign( </code><var>D</var><code>, </code><var>x</var><code> )</code>
	334	<p>
	335	Suppose <var>D</var> is a block design, and <var>x</var> is a point or block of <var>D</var>.
	336	Then this function returns the <strong>derived design</strong>
	337	<a name = "I0"></a>
	338
	339	<i>DD</i> of <var>D</var>, with respect to <var>x</var>.
	340	<p>
	341	If <var>x</var> is a point then <i>DD</i> is the block design whose blocks are those
	342	of <var>D</var> containing <var>x</var>, but with <var>x</var> deleted from these blocks, and the
	343	points of <i>DD</i> are those which occur in some block of <i>DD</i>.
	344	<p>
	345	If <var>x</var> is a block, then the points of <i>DD</i> are the points in <var>x</var>, and
	346	the blocks of <i>DD</i> are the blocks of <var>D</var> other than <var>x</var> containing at
	347	least one point of <var>x</var>, but with all points not in <var>x</var> deleted from
	348	these blocks. Note that any repeat of <var>x</var>, but not <var>x</var> itself, is a
	349	block of <i>DD</i>.
	350	<p>
	351	It is an error if the resulting block design <i>DD</i> has no blocks or an
	352	empty block.
	353	<p>
	354	The points of <i>DD</i> are relabelled 1,2,·.·, preserving the order of
	355	the corresponding points of <var>D</var>; the point-names of <i>DD</i> (listed in
	356	<code><i>DD</i>.pointNames</code>) are those of these corresponding points of <var>D</var>.
	357	<p>
	358	<pre>
	359	gap> D:=BlockDesigns(rec(v:=11,blockSizes:=[5],
	360	> tSubsetStructure:=rec(t:=2,lambdas:=[2])))[1];
	361	rec( isBlockDesign := true, v := 11,
	362	blocks := [ [ 1, 2, 3, 4, 5 ], [ 1, 2, 9, 10, 11 ], [ 1, 3, 6, 7, 9 ],
	363	[ 1, 4, 7, 8, 10 ], [ 1, 5, 6, 8, 11 ], [ 2, 3, 6, 8, 10 ],
	364	[ 2, 4, 6, 7, 11 ], [ 2, 5, 7, 8, 9 ], [ 3, 4, 8, 9, 11 ],
	365	[ 3, 5, 7, 10, 11 ], [ 4, 5, 6, 9, 10 ] ],
	366	tSubsetStructure := rec( t := 2, lambdas := [ 2 ] ), isBinary := true,
	367	isSimple := true, blockSizes := [ 5 ], blockNumbers := [ 11 ], r := 5,
	368	autGroup := Group([ (2,4)(3,5)(7,11)(8,9), (1,3)(2,5)(7,9)(10,11),
	369	(1,5,3)(6,11,7)(8,10,9), (1,10,5,2,11,3)(4,9,7)(6,8) ]) )
	370	gap> AllTDesignLambdas(D);
	371	[ 11, 5, 2 ]
	372	gap> DD:=DerivedBlockDesign(D,6);
	373	rec( isBlockDesign := true, v := 10,
	374	blocks := [ [ 1, 3, 6, 8 ], [ 1, 5, 7, 10 ], [ 2, 3, 7, 9 ],
	375	[ 2, 4, 6, 10 ], [ 4, 5, 8, 9 ] ],
	376	pointNames := [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 11 ] )
	377	gap> AllTDesignLambdas(DD);
	378	[ 5, 2 ]
	379	gap> DD:=DerivedBlockDesign(D,D.blocks[6]);
	380	rec( isBlockDesign := true, v := 5,
	381	blocks := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 3 ], [ 2, 4 ],
	382	[ 2, 5 ], [ 3, 4 ], [ 3, 5 ], [ 4, 5 ] ],
	383	pointNames := [ 2, 3, 6, 8, 10 ] )
	384	gap> AllTDesignLambdas(DD);
	385	[ 10, 4, 1 ]
	386	</pre>
	387	<p>
	388	<a name = "SSEC001.12"></a>
	389	<li><code>ResidualBlockDesign( </code><var>D</var><code>, </code><var>x</var><code> )</code>
	390	<p>
	391	Suppose <var>D</var> is a block design, and <var>x</var> is a point or block of <var>D</var>.
	392	Then this function returns the <strong>residual design</strong>
	393	<a name = "I1"></a>
	394
	395	<i>RD</i> of <var>D</var>, with respect to <var>x</var>.
	396	<p>
	397	If <var>x</var> is a point then <i>RD</i> is the block design whose blocks are those
	398	of <var>D</var> not containing <var>x</var>, and the points of <i>RD</i> are those which occur
	399	in some block of <i>RD</i>.
	400	<p>
	401	If <var>x</var> is a block, then the points of <i>RD</i> are those of <var>D</var> not in
	402	<var>x</var>, and the blocks of <i>RD</i> are the blocks of <var>D</var> (including repeats)
	403	containing at least one point not in <var>x</var>, but with all points in <var>x</var>
	404	deleted from these blocks.
	405	<p>
	406	It is an error if the resulting block design <i>RD</i> has no blocks.
	407	<p>
	408	The points of <i>RD</i> are relabelled 1,2,·.·, preserving the order of
	409	the corresponding points of <var>D</var>; the point-names of <i>RD</i> (listed in
	410	<code><i>RD</i>.pointNames</code>) are those of these corresponding points of <var>D</var>.
	411	<p>
	412	<pre>
	413	gap> D:=BlockDesigns(rec(v:=11,blockSizes:=[5],
	414	> tSubsetStructure:=rec(t:=2,lambdas:=[2])))[1];
	415	rec( isBlockDesign := true, v := 11,
	416	blocks := [ [ 1, 2, 3, 4, 5 ], [ 1, 2, 9, 10, 11 ], [ 1, 3, 6, 7, 9 ],
	417	[ 1, 4, 7, 8, 10 ], [ 1, 5, 6, 8, 11 ], [ 2, 3, 6, 8, 10 ],
	418	[ 2, 4, 6, 7, 11 ], [ 2, 5, 7, 8, 9 ], [ 3, 4, 8, 9, 11 ],
	419	[ 3, 5, 7, 10, 11 ], [ 4, 5, 6, 9, 10 ] ],
	420	tSubsetStructure := rec( t := 2, lambdas := [ 2 ] ), isBinary := true,
	421	isSimple := true, blockSizes := [ 5 ], blockNumbers := [ 11 ], r := 5,
	422	autGroup := Group([ (2,4)(3,5)(7,11)(8,9), (1,3)(2,5)(7,9)(10,11),
	423	(1,5,3)(6,11,7)(8,10,9), (1,10,5,2,11,3)(4,9,7)(6,8) ]) )
	424	gap> AllTDesignLambdas(D);
	425	[ 11, 5, 2 ]
	426	gap> RD:=ResidualBlockDesign(D,6);
	427	rec( isBlockDesign := true, v := 10,
	428	blocks := [ [ 1, 2, 3, 4, 5 ], [ 1, 2, 8, 9, 10 ], [ 1, 4, 6, 7, 9 ],
	429	[ 2, 5, 6, 7, 8 ], [ 3, 4, 7, 8, 10 ], [ 3, 5, 6, 9, 10 ] ],
	430	pointNames := [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 11 ] )
	431	gap> AllTDesignLambdas(RD);
	432	[ 6, 3 ]
	433	gap> RD:=ResidualBlockDesign(D,D.blocks[6]);
	434	rec( isBlockDesign := true, v := 6,
	435	blocks := [ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 6 ], [ 1, 4, 5 ],
	436	[ 1, 5, 6 ], [ 2, 3, 5 ], [ 2, 4, 6 ], [ 2, 5, 6 ], [ 3, 4, 5 ],
	437	[ 3, 4, 6 ] ], pointNames := [ 1, 4, 5, 7, 9, 11 ] )
	438	gap> AllTDesignLambdas(RD);
	439	[ 10, 5, 2 ]
	440	</pre>
	441	<p>
	442	<a name = "SSEC001.13"></a>
	443	<li><code>TDesignFromTBD( </code><var>D</var><code>, </code><var>t</var><code>, </code><var>k</var><code> )</code>
	444	<p>
	445	For <i>t</i> a non-negative integer, <i>K</i> a set of positive integers, and
	446	<i>v</i>,λ positive integers with <i>t</i> ≤ <i>v</i>, a <i>t</i>-<strong>wise balanced design</strong>,
	447	or a <i>t</i>-(<i>v</i>,<i>K</i>,λ) <strong>design</strong>, is a binary block design with exactly
	448	<i>v</i> points, such that each block has size in <i>K</i> and each <i>t</i>-subset of
	449	the points is contained in exactly λ blocks.
	450	<p>
	451	Now let <var>t</var> and <var>k</var> be positive integers, <var>D</var> be a <var>t</var>-(<i>v</i>,<i>K</i>,λ)
	452	design (for some set <i>K</i>), and <i>t</i> ≤ <i>k</i> ≤ <i>k</i><sub>1</sub>, where exactly <i>s</i>
	453	distinct block-sizes <i>k</i><sub>1</sub> < … < <i>k</i><sub><i>s</i></sub> occur in <var>D</var>. Then this
	454	function returns the <var>t</var>-design <i>D</i><sup></sup>=<i>D</i><sup></sup>(<i>t</i>,<i>k</i>) described and studied
	455	in <a href="biblio.htm#McSo"><cite>McSo</cite></a>.
	456	<p>
	457	The point set of <i>D</i><sup></sup> is that of <var>D</var>, and the block multiset of <i>D</i><sup></sup>
	458	consists of, for each <i>i</i>=1,…,<i>s</i> and each block <i>B</i> of <var>D</var> of size
	459	<i>k</i><sub><i>i</i></sub> (including repeats), exactly <br clear="all" /><table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"><i>n</i>/</td><td align="left" class="cl">⎛<br />⎝</td><td nowrap="nowrap" align="center"><i>k</i><sub><i>i</i></sub>−<i>t</i><br /><i>k</i>−<i>t</i><br /></td><td align="left" class="cl">⎞<br />⎠</td><td nowrap="nowrap" align="center"></td></tr></table></td></tr></table> copies of
	460	every <i>k</i>-subset of <i>B</i>, where <br clear="all" /><table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"><i>n</i>:=\lcm(</td><td align="left" class="cl">⎛<br />⎝</td><td nowrap="nowrap" align="center"><i>k</i><sub><i>i</i></sub>−<i>t</i><br /><i>k</i>−<i>t</i><br /></td><td align="left" class="cl">⎞<br />⎠</td><td nowrap="nowrap" align="center">:1 ≤ <i>i</i> ≤ <i>s</i>)</td></tr></table></td></tr></table>.
	461	<p>
	462	It is shown in <a href="biblio.htm#McSo"><cite>McSo</cite></a> that <i>D</i><sup>*</sup> is a <i>t</i>-(<i>v</i>,<i>k</i>,<i>n</i>λ) design,
	463	that \Aut(<i>D</i>) ⊆ \Aut(<i>D</i><sup>*</sup>), and that if λ = 1 and <i>t</i> < <i>k</i>,
	464	then \Aut(<i>D</i>)=\Aut(<i>D</i><sup>*</sup>).
	465	<p>
	466	<pre>
	467	gap> D:=BlockDesigns(rec(v:=10, blockSizes:=[3,4],
	468	> tSubsetStructure:=rec(t:=2,lambdas:=[1])))[1];
	469	rec( isBlockDesign := true, v := 10,
	470	blocks := [ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 10 ],
	471	[ 2, 6, 8 ], [ 2, 7, 9 ], [ 3, 5, 9 ], [ 3, 6, 10 ], [ 3, 7, 8 ],
	472	[ 4, 5, 8 ], [ 4, 6, 9 ], [ 4, 7, 10 ] ],
	473	tSubsetStructure := rec( t := 2, lambdas := [ 1 ] ), isBinary := true,
	474	isSimple := true, blockSizes := [ 3, 4 ], blockNumbers := [ 9, 3 ],
	475	autGroup := Group([ (5,6,7)(8,9,10), (2,3)(5,7)(8,10),
	476	(2,3,4)(5,7,6)(8,9,10), (2,3,4)(5,9,6,8,7,10), (2,6,9,3,7,10)(4,5,8) ])
	477	)
	478	gap> PairwiseBalancedLambda(D);
	479	1
	480	gap> Dstar:=TDesignFromTBD(D,2,3);
	481	rec( isBlockDesign := true, v := 10,
	482	blocks := [ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 4 ], [ 1, 5, 6 ],
	483	[ 1, 5, 7 ], [ 1, 6, 7 ], [ 1, 8, 9 ], [ 1, 8, 10 ], [ 1, 9, 10 ],
	484	[ 2, 3, 4 ], [ 2, 5, 10 ], [ 2, 5, 10 ], [ 2, 6, 8 ], [ 2, 6, 8 ],
	485	[ 2, 7, 9 ], [ 2, 7, 9 ], [ 3, 5, 9 ], [ 3, 5, 9 ], [ 3, 6, 10 ],
	486	[ 3, 6, 10 ], [ 3, 7, 8 ], [ 3, 7, 8 ], [ 4, 5, 8 ], [ 4, 5, 8 ],
	487	[ 4, 6, 9 ], [ 4, 6, 9 ], [ 4, 7, 10 ], [ 4, 7, 10 ], [ 5, 6, 7 ],
	488	[ 8, 9, 10 ] ],
	489	autGroup := Group([ (5,6,7)(8,9,10), (2,3)(5,7)(8,10), (2,3,4)(5,7,6)(8,9,
	490	10), (2,3,4)(5,9,6,8,7,10), (2,6,9,3,7,10)(4,5,8) ]) )
	491	gap> AllTDesignLambdas(Dstar);
	492	[ 30, 9, 2 ]
	493	</pre>
	494	<p>
	495	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
	496	<P>
	497	<address>design manual<br>March 2019
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	0	<html><head><title>[design] 4 Determining basic properties of block designs</title></head>
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	3	<h1>4 Determining basic properties of block designs</h1><p>
	4	<P>
	5	<H3>Sections</H3>
	6	<oL>
	7	<li> <A HREF="CHAP004.htm#SECT001">The functions for basic properties</a>
	8	</ol><p>
	9	<p>
	10	<p>
	11	<h2><a name="SECT001">4.1 The functions for basic properties</a></h2>
	12	<p><p>
	13	<a name = "SSEC001.1"></a>
	14	<li><code>IsBlockDesign( </code><var>obj</var><code> )</code>
	15	<p>
	16	This boolean function returns <code>true</code> if and only if <var>obj</var>, which can be
	17	an object of arbitrary type, is a block design.
	18	<p>
	19	<pre>
	20	gap> IsBlockDesign(5);
	21	false
	22	gap> IsBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) );
	23	true
	24	</pre>
	25	<p>
	26	<a name = "SSEC001.2"></a>
	27	<li><code>IsBinaryBlockDesign( </code><var>D</var><code> )</code>
	28	<p>
	29	This boolean function returns <code>true</code> if and only if the block design
	30	<var>D</var> is <strong>binary</strong>, that is, if no block of <var>D</var> has a repeated element.
	31	<p>
	32	<pre>
	33	gap> IsBinaryBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) );
	34	true
	35	gap> IsBinaryBlockDesign( BlockDesign(2,[[1],[1,2],[1,2,2]]) );
	36	false
	37	</pre>
	38	<p>
	39	<a name = "SSEC001.3"></a>
	40	<li><code>IsSimpleBlockDesign( </code><var>D</var><code> )</code>
	41	<p>
	42	This boolean function returns <code>true</code> if and only if the block design
	43	<var>D</var> is <strong>simple</strong>, that is, if no block of <var>D</var> is repeated.
	44	<p>
	45	<pre>
	46	gap> IsSimpleBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) );
	47	false
	48	gap> IsSimpleBlockDesign( BlockDesign(2,[[1],[1,2],[1,2,2]]) );
	49	true
	50	</pre>
	51	<p>
	52	<a name = "SSEC001.4"></a>
	53	<li><code>IsConnectedBlockDesign( </code><var>D</var><code> )</code>
	54	<p>
	55	This boolean function returns <code>true</code> if and only if the block design
	56	<var>D</var> is <strong>connected</strong>, that is, if its incidence graph is a connected
	57	graph.
	58	<p>
	59	<pre>
	60	gap> IsConnectedBlockDesign( BlockDesign(2,[[1],[2]]) );
	61	false
	62	gap> IsConnectedBlockDesign( BlockDesign(2,[[1,2]]) );
	63	true
	64	</pre>
	65	<p>
	66	<a name = "SSEC001.5"></a>
	67	<li><code>BlockDesignPoints( </code><var>D</var><code> )</code>
	68	<p>
	69	This function returns the set of points of the block design <var>D</var>, that is
	70	<code>[1..</code><var>D</var><code>.v]</code>. The returned result is immutable.
	71	<p>
	72	<pre>
	73	gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]);
	74	rec( isBlockDesign := true, v := 3,
	75	blocks := [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] )
	76	gap> BlockDesignPoints(D);
	77	[ 1 .. 3 ]
	78	</pre>
	79	<p>
	80	<a name = "SSEC001.6"></a>
	81	<li><code>NrBlockDesignPoints( </code><var>D</var><code> )</code>
	82	<p>
	83	This function returns the number of points of the block design <var>D</var>.
	84	<p>
	85	<pre>
	86	gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]);
	87	rec( isBlockDesign := true, v := 3,
	88	blocks := [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] )
	89	gap> NrBlockDesignPoints(D);
	90	3
	91	</pre>
	92	<p>
	93	<a name = "SSEC001.7"></a>
	94	<li><code>BlockDesignBlocks( </code><var>D</var><code> )</code>
	95	<p>
	96	This function returns the (sorted) list of blocks of the block design <var>D</var>.
	97	The returned result is immutable.
	98	<p>
	99	<pre>
	100	gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]);
	101	rec( isBlockDesign := true, v := 3,
	102	blocks := [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] )
	103	gap> BlockDesignBlocks(D);
	104	[ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ]
	105	</pre>
	106	<p>
	107	<a name = "SSEC001.8"></a>
	108	<li><code>NrBlockDesignBlocks( </code><var>D</var><code> )</code>
	109	<p>
	110	This function returns the number of blocks of the block design <var>D</var>.
	111	<p>
	112	<pre>
	113	gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]);
	114	rec( isBlockDesign := true, v := 3,
	115	blocks := [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] )
	116	gap> NrBlockDesignBlocks(D);
	117	4
	118	</pre>
	119	<p>
	120	<a name = "SSEC001.9"></a>
	121	<li><code>BlockSizes( </code><var>D</var><code> )</code>
	122	<p>
	123	This function returns the set of sizes (actually list-lengths) of the
	124	blocks of the block design <var>D</var>.
	125	<p>
	126	<pre>
	127	gap> BlockSizes( BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]) );
	128	[ 1, 3 ]
	129	</pre>
	130	<p>
	131	<a name = "SSEC001.10"></a>
	132	<li><code>BlockNumbers( </code><var>D</var><code> )</code>
	133	<p>
	134	Let <var>D</var> be a block design. Then this function returns a list of
	135	the same length as <code>BlockSizes(</code><var>D</var><code>)</code>, such that the <i>i</i>-th element
	136	of this returned list is the number of blocks of <var>D</var> of size
	137	<code>BlockSizes(</code><var>D</var><code>)[<i>i</i>]</code>.
	138	<p>
	139	<pre>
	140	gap> D:=BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]);
	141	rec( isBlockDesign := true, v := 3,
	142	blocks := [ [ 1 ], [ 1, 2, 2 ], [ 1, 2, 3 ], [ 2 ], [ 3 ] ] )
	143	gap> BlockSizes(D);
	144	[ 1, 3 ]
	145	gap> BlockNumbers(D);
	146	[ 3, 2 ]
	147	</pre>
	148	<p>
	149	<a name = "SSEC001.11"></a>
	150	<li><code>ReplicationNumber( </code><var>D</var><code> )</code>
	151	<p>
	152	If the block design <var>D</var> is equireplicate, then this function returns
	153	its replication number; otherwise <code>fail</code> is returned.
	154	<p>
	155	A block design <i>D</i> is <strong>equireplicate</strong> with <strong>replication number</strong> <i>r</i> if,
	156	for every point <i>x</i> of <i>D</i>, <i>r</i> is equal to the sum over the blocks of
	157	the multiplicity of <i>x</i> in a block. For a binary block design this is
	158	the same as saying that each point <i>x</i> is contained in exactly <i>r</i> blocks.
	159	<p>
	160	<pre>
	161	gap> ReplicationNumber(BlockDesign(4,[[1],[1,2],[2,3,3],[4,4]]));
	162	2
	163	gap> ReplicationNumber(BlockDesign(4,[[1],[1,2],[2,3],[4,4]]));
	164	fail
	165	</pre>
	166	<p>
	167	<a name = "SSEC001.12"></a>
	168	<li><code>PairwiseBalancedLambda( </code><var>D</var><code> )</code>
	169	<p>
	170	A binary block design <i>D</i> is <strong>pairwise balanced</strong> if <i>D</i> has at least two
	171	points and every pair of distinct points is contained in exactly λ
	172	blocks, for some positive constant λ.
	173	<p>
	174	Given a binary block design <var>D</var>, this function returns <code>fail</code> if <var>D</var> is
	175	not pairwise balanced, and otherwise the positive constant λ such
	176	that every pair of distinct points of <var>D</var> is in exactly λ blocks.
	177	<p>
	178	<pre>
	179	gap> D:=BlockDesigns(rec(v:=10, blockSizes:=[3,4],
	180	> tSubsetStructure:=rec(t:=2,lambdas:=[1])))[1];
	181	rec( isBlockDesign := true, v := 10,
	182	blocks := [ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 10 ],
	183	[ 2, 6, 8 ], [ 2, 7, 9 ], [ 3, 5, 9 ], [ 3, 6, 10 ], [ 3, 7, 8 ],
	184	[ 4, 5, 8 ], [ 4, 6, 9 ], [ 4, 7, 10 ] ],
	185	tSubsetStructure := rec( t := 2, lambdas := [ 1 ] ), isBinary := true,
	186	isSimple := true, blockSizes := [ 3, 4 ], blockNumbers := [ 9, 3 ],
	187	autGroup := Group([ (5,6,7)(8,9,10), (2,3)(5,7)(8,10),
	188	(2,3,4)(5,7,6)(8,9,10), (2,3,4)(5,9,6,8,7,10), (2,6,9,3,7,10)(4,5,8) ])
	189	)
	190	gap> PairwiseBalancedLambda(D);
	191	1
	192	</pre>
	193	<p>
	194	<a name = "SSEC001.13"></a>
	195	<li><code>TSubsetLambdasVector( </code><var>D</var><code>, </code><var>t</var><code> )</code>
	196	<p>
	197	Let <var>D</var> be a block design, <var>t</var> a non-negative integer, and
	198	<code><i>v</i>=</code><var>D</var><code>.v</code>. Then this function returns an integer vector <i>L</i>
	199	whose positions correspond to the <var>t</var>-subsets of {1,…,<i>v</i>}.
	200	The <i>i</i>-th element of <i>L</i> is the sum over all blocks <i>B</i> of <var>D</var>
	201	of the number of times the <i>i</i>-th <var>t</var>-subset (in lexicographic order)
	202	is contained in <i>B</i>. (For example, if <i>t</i>=2 and <i>B</i>=[1,1,2,3,3,4], then
	203	<i>B</i> contains [1,2] twice, [1,3] four times, [1,4] twice,
	204	[2,3] twice, [2,4] once, and [3,4] twice.) In particular,
	205	if <var>D</var> is binary then <i>L</i>[<i>i</i>] is simply the number of blocks of <var>D</var>
	206	containing the <i>i</i>-th <var>t</var>-subset (in lexicographic order).
	207	<p>
	208	<pre>
	209	gap> D:=BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]);;
	210	gap> TSubsetLambdasVector(D,0);
	211	[ 5 ]
	212	gap> TSubsetLambdasVector(D,1);
	213	[ 3, 4, 2 ]
	214	gap> TSubsetLambdasVector(D,2);
	215	[ 3, 1, 1 ]
	216	gap> TSubsetLambdasVector(D,3);
	217	[ 1 ]
	218	</pre>
	219	<p>
	220	<a name = "SSEC001.14"></a>
	221	<li><code>AllTDesignLambdas( </code><var>D</var><code> )</code>
	222	<p>
	223	If the block design <var>D</var> is not a <i>t</i>-design for some <i>t</i> ≥ 0 then this
	224	function returns an empty list. Otherwise <var>D</var> is a binary block design
	225	with constant block size <i>k</i>, say, and this function returns a list
	226	<i>L</i> of length <i>T</i>+1, where <i>T</i> is the maximum <i>t</i> ≤ <i>k</i> such that <var>D</var>
	227	is a <i>t</i>-design, and, for <i>i</i>=1,…,<i>T</i>+1, <i>L</i>[<i>i</i>] is equal to the
	228	(constant) number of blocks of <var>D</var> containing an (<i>i</i>−1)-subset of
	229	the point-set of <var>D</var>. The returned result is immutable.
	230	<p>
	231	<pre>
	232	gap> AllTDesignLambdas(PGPointFlatBlockDesign(3,2,1));
	233	[ 35, 7, 1 ]
	234	</pre>
	235	<p>
	236	<a name = "SSEC001.15"></a>
	237	<li><code>AffineResolvableMu( </code><var>D</var><code> )</code>
	238	<p>
	239	A block design is <strong>affine resolvable</strong> if the design is resolvable
	240	and any two blocks not in the same parallel class of a resolution
	241	meet in a constant number μ of points.
	242	<p>
	243	If the block design <var>D</var> is affine resolvable, then this function
	244	returns its value of μ; otherwise <code>fail</code> is returned.
	245	<p>
	246	The value 0 is returned if, and only if, <var>D</var> consists of a single
	247	parallel class.
	248	<p>
	249	<pre>
	250	gap> P:=PGPointFlatBlockDesign(2,3,1);; # projective plane of order 3
	251	gap> AffineResolvableMu(P);
	252	fail
	253	gap> A:=ResidualBlockDesign(P,P.blocks[1]);; # affine plane of order 3
	254	gap> AffineResolvableMu(A);
	255	1
	256	</pre>
	257	<p>
	258	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP003.htm">Previous</a>] [<a href ="CHAP005.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
	259	<P>
	260	<address>design manual<br>March 2019
	261	</address></body></html>⏎

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	1	<body text="#000000" bgcolor="#ffffff">
	2	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href ="CHAP006.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
	3	<h1>5 Matrices and efficiency measures for block designs</h1><p>
	4	<P>
	5	<H3>Sections</H3>
	6	<oL>
	7	<li> <A HREF="CHAP005.htm#SECT001">Matrices associated with a block design</a>
	8	<li> <A HREF="CHAP005.htm#SECT002">The function BlockDesignEfficiency</a>
	9	<li> <A HREF="CHAP005.htm#SECT003">Computing an interval for a certain real zero of a rational polynomial</a>
	10	</ol><p>
	11	<p>
	12	In this chapter we describe functions to calculate certain matrices
	13	associated with a block design, and the function <code>BlockDesignEfficiency</code>
	14	which determines certain statistical efficiency measures of a 1-design.
	15	<p>
	16	We also document the utility function <code>DESIGN_IntervalForLeastRealZero</code>,
	17	which is used in the calculation of E-efficiency measures, but has much
	18	wider application.
	19	<p>
	20	<p>
	21	<h2><a name="SECT001">5.1 Matrices associated with a block design</a></h2>
	22	<p><p>
	23	<a name = "SSEC001.1"></a>
	24	<li><code>PointBlockIncidenceMatrix( </code><var>D</var><code> )</code>
	25	<p>
	26	This function returns the point-block incidence matrix <var>N</var> of the
	27	block design <var>D</var>. This matrix has rows indexed by the points of <var>D</var>
	28	and columns by the blocks of <var>D</var>, with the (<i>i</i>,<i>j</i>)-entry of <var>N</var> being
	29	the number of times point <i>i</i> occurs in <code></code><var>D</var><code>.blocks[</code><var>j</var><code>]</code>.
	30	<p>
	31	The returned matrix <var>N</var> is immutable.
	32	<p>
	33	<pre>
	34	gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));;
	35	gap> BlockDesignBlocks(D);
	36	[ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 8, 11 ],
	37	[ 2, 7, 9, 12 ], [ 3, 5, 10, 12 ], [ 3, 6, 9, 11 ], [ 4, 6, 8, 12 ],
	38	[ 4, 7, 10, 11 ] ]
	39	gap> PointBlockIncidenceMatrix(D);
	40	[ [ 1, 1, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 1, 1, 0, 0, 0, 0 ],
	41	[ 1, 0, 0, 0, 0, 1, 1, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 1, 1 ],
	42	[ 0, 1, 0, 1, 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 1, 1, 0 ],
	43	[ 0, 1, 0, 0, 1, 0, 0, 0, 1 ], [ 0, 0, 1, 1, 0, 0, 0, 1, 0 ],
	44	[ 0, 0, 1, 0, 1, 0, 1, 0, 0 ], [ 0, 0, 1, 0, 0, 1, 0, 0, 1 ],
	45	[ 0, 0, 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 0, 1, 1, 0, 1, 0 ] ]
	46	</pre>
	47	<p>
	48	<a name = "SSEC001.2"></a>
	49	<li><code>ConcurrenceMatrix( </code><var>D</var><code> )</code>
	50	<p>
	51	This function returns the concurrence matrix <var>L</var> of the block design <var>D</var>.
	52	This matrix is equal to <i>NN</i><sup><span class="roman">T</span></sup>, where <i>N</i> is the point-block
	53	incidence matrix of <var>D</var> (see <a href="CHAP005.htm#SSEC001.1">PointBlockIncidenceMatrix</a>) and
	54	<i>N</i><sup><span class="roman">T</span></sup> is the transpose of <i>N</i>. If <var>D</var> is a binary block design
	55	then the (<i>i</i>,<i>j</i>)-entry of its concurrence matrix is the number of blocks
	56	containing points <i>i</i> and <i>j</i>.
	57	<p>
	58	The returned matrix <var>L</var> is immutable.
	59	<p>
	60	<pre>
	61	gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));;
	62	gap> BlockDesignBlocks(D);
	63	[ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 8, 11 ],
	64	[ 2, 7, 9, 12 ], [ 3, 5, 10, 12 ], [ 3, 6, 9, 11 ], [ 4, 6, 8, 12 ],
	65	[ 4, 7, 10, 11 ] ]
	66	gap> ConcurrenceMatrix(D);
	67	[ [ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0 ],
	68	[ 1, 3, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1 ],
	69	[ 1, 1, 3, 1, 1, 1, 0, 0, 1, 1, 1, 1 ],
	70	[ 1, 1, 1, 3, 0, 1, 1, 1, 0, 1, 1, 1 ],
	71	[ 1, 1, 1, 0, 3, 1, 1, 1, 0, 1, 1, 1 ],
	72	[ 1, 0, 1, 1, 1, 3, 1, 1, 1, 0, 1, 1 ],
	73	[ 1, 1, 0, 1, 1, 1, 3, 0, 1, 1, 1, 1 ],
	74	[ 1, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1 ],
	75	[ 1, 1, 1, 0, 0, 1, 1, 1, 3, 1, 1, 1 ],
	76	[ 1, 0, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1 ],
	77	[ 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 0 ],
	78	[ 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3 ] ]
	79	</pre>
	80	<p>
	81	<a name = "SSEC001.3"></a>
	82	<li><code>InformationMatrix( </code><var>D</var><code> )</code>
	83	<p>
	84	This function returns the information matrix <var>C</var> of the block design <var>D</var>.
	85	<p>
	86	This matrix is defined as follows. Suppose <var>D</var> has <i>v</i> points and <i>b</i>
	87	blocks, let <i>R</i> be the <i>v</i>×<i>v</i> diagonal matrix whose (<i>i</i>,<i>i</i>)-entry
	88	is the replication number of the point <i>i</i>, let <i>N</i> be the point-block
	89	incidence matrix of <var>D</var> (see <a href="CHAP005.htm#SSEC001.1">PointBlockIncidenceMatrix</a>), and let <i>K</i>
	90	be the <i>b</i>×<i>b</i> diagonal matrix whose (<i>j</i>,<i>j</i>)-entry is the length of
	91	<code></code><var>D</var><code>.blocks[</code><var>j</var><code>]</code>. Then the <strong>information matrix</strong> of <var>D</var> is
	92	<i>C</i>:=<i>R</i>−<i>NK</i><sup>−1</sup><i>N</i><sup><span class="roman">T</span></sup>. If <var>D</var> is a 1-(<i>v</i>,<i>k</i>,<i>r</i>) design then this expression
	93	for <var>C</var> simplifies to <i>rI</i>−<i>k</i><sup>−1</sup><i>L</i>, where <i>I</i> is the <i>v</i>×<i>v</i> identity
	94	matrix and <i>L</i> is the concurrence matrix of <var>D</var> (see <a href="CHAP005.htm#SSEC001.2">ConcurrenceMatrix</a>).
	95	<p>
	96	The returned matrix <var>C</var> is immutable.
	97	<p>
	98	<pre>
	99	gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));;
	100	gap> BlockDesignBlocks(D);
	101	[ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 8, 11 ],
	102	[ 2, 7, 9, 12 ], [ 3, 5, 10, 12 ], [ 3, 6, 9, 11 ], [ 4, 6, 8, 12 ],
	103	[ 4, 7, 10, 11 ] ]
	104	gap> InformationMatrix(D);
	105	[ [ 9/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, 0, 0 ],
	106	[ -1/4, 9/4, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4, 0, -1/4, -1/4 ],
	107	[ -1/4, -1/4, 9/4, -1/4, -1/4, -1/4, 0, 0, -1/4, -1/4, -1/4, -1/4 ],
	108	[ -1/4, -1/4, -1/4, 9/4, 0, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4 ],
	109	[ -1/4, -1/4, -1/4, 0, 9/4, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4 ],
	110	[ -1/4, 0, -1/4, -1/4, -1/4, 9/4, -1/4, -1/4, -1/4, 0, -1/4, -1/4 ],
	111	[ -1/4, -1/4, 0, -1/4, -1/4, -1/4, 9/4, 0, -1/4, -1/4, -1/4, -1/4 ],
	112	[ -1/4, -1/4, 0, -1/4, -1/4, -1/4, 0, 9/4, -1/4, -1/4, -1/4, -1/4 ],
	113	[ -1/4, -1/4, -1/4, 0, 0, -1/4, -1/4, -1/4, 9/4, -1/4, -1/4, -1/4 ],
	114	[ -1/4, 0, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4, 9/4, -1/4, -1/4 ],
	115	[ 0, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, 9/4, 0 ],
	116	[ 0, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, 0, 9/4 ] ]
	117	</pre>
	118	<p>
	119	<p>
	120	<h2><a name="SECT002">5.2 The function BlockDesignEfficiency</a></h2>
	121	<p><p>
	122	<a name = "SSEC002.1"></a>
	123	<li><code>BlockDesignEfficiency( </code><var>D</var><code> )</code>
	124	<li><code>BlockDesignEfficiency( </code><var>D</var><code>, </code><var>eps</var><code> )</code>
	125	<li><code>BlockDesignEfficiency( </code><var>D</var><code>, </code><var>eps</var><code>, </code><var>includeMV</var><code> )</code>
	126	<p>
	127	Let <var>D</var> be a 1-(<i>v</i>,<i>k</i>,<i>r</i>) design with <i>v</i> > 1, let <var>eps</var> be a positive
	128	rational number (default: 10<sup>−6</sup>), and let <var>includeMV</var> be a boolean
	129	(default: <code>false</code>). Then this function returns a record <var>eff</var> containing
	130	information on statistical efficiency measures of <var>D</var>. These measures
	131	are defined below. See <a href="biblio.htm#Extrep"><cite>Extrep</cite></a>, <a href="biblio.htm#BaCa"><cite>BaCa</cite></a> and <a href="biblio.htm#BaRo"><cite>BaRo</cite></a>
	132	for further details. All returned results are computed using exact
	133	algebraic computation.
	134	<p>
	135	The component <code></code><var>eff</var><code>.A</code> contains the A-efficiency measure for <var>D</var>,
	136	<code></code><var>eff</var><code>.Dpowered</code> contains the D-efficiency measure of <var>D</var> raised to the
	137	power <i>v</i>−1, and <code></code><var>eff</var><code>.Einterval</code> is a list [<i>a</i>,<i>b</i>] of non-negative
	138	rational numbers such that if <i>x</i> is the E-efficiency measure of <var>D</var>
	139	then <i>a</i> ≤ <i>x</i> ≤ <i>b</i>, <i>b</i>−<i>a</i> ≤ <var>eps</var>, and if <i>x</i> is rational then <i>a</i>=<i>x</i>=<i>b</i>.
	140	Moreover <code></code><var>eff</var><code>.CEFpolynomial</code> contains the monic polynomial over the
	141	rationals whose zeros (counting multiplicities) are the canonical
	142	efficiency factors of the design <var>D</var>. If <code></code><var>includeMV</var><code>=true</code> then
	143	additional work is done to compute the MV- (also called E′-) efficiency
	144	measure, and then <code></code><var>eff</var><code>.MV</code> contains the value of this measure. (This
	145	component may be set even if <code></code><var>includeMV</var><code>=false</code>, as a byproduct of
	146	other computation.)
	147	<p>
	148	We now define the canonical efficiency factors and the A-, D-, E-,
	149	and MV-efficiency measures of a 1-design.
	150	<p>
	151	Let <i>D</i> be a 1-(<i>v</i>,<i>k</i>,<i>r</i>) design with <i>v</i> ≥ 2, let <i>C</i> be the information
	152	matrix of <i>D</i> (see <a href="CHAP005.htm#SSEC001.3">InformationMatrix</a>), and let <i>F</i>:=<i>r</i><sup>−1</sup><i>C</i>.
	153	The eigenvalues of <i>F</i> are all real and lie in the interval [0,1].
	154	At least one of these eigenvalues is zero: an associated eigenvector is
	155	the all-1 vector. The remaining eigenvalues δ<sub>1</sub> ≤ δ<sub>2</sub> ≤ … ≤ δ<sub><i>v</i>−1</sub> of <i>F</i> are called the <strong>canonical efficiency
	156	factors</strong> of <i>D</i>. These are all non-zero if and only if <i>D</i> is connected
	157	(that is, the point-block incidence graph of <i>D</i> is a connected graph).
	158	<p>
	159	If <i>D</i> is not connected, then the A-, D-, E-, and MV-efficiency measures
	160	of <i>D</i> are all defined to be zero. Otherwise, the <strong>A-efficiency
	161	measure</strong> is (<i>v</i>−1)/∑<sub><i>i</i>=1</sub><sup><i>v</i>−1</sup>1/δ<sub><i>i</i></sub> (the harmonic mean
	162	of the canonical efficiency factors), the <strong>D-efficiency measure</strong>
	163	is (∏<sub><i>i</i>=1</sub><sup><i>v</i>−1</sup>δ<sub><i>i</i></sub>)<sup>1/(<i>v</i>−1)</sup> (the geometric mean of
	164	the canonical efficiency factors), and the <strong>E-efficiency measure</strong> is
	165	δ<sub>1</sub> (the minimum of the canonical efficiency factors).
	166	<p>
	167	If <i>D</i> is connected, and the MV-efficiency measure is required,
	168	then it is computed as follows. Let <i>F</i>:=<i>r</i><sup>−1</sup><i>C</i> be as before,
	169	and let <i>P</i>:=<i>v</i><sup>−1</sup><i>J</i>, where <i>J</i> is the <i>v</i>×<i>v</i> all-1 matrix. Set
	170	<i>M</i>:=(<i>F</i>+<i>P</i>)<sup>−1</sup>−<i>P</i>, making <i>M</i> the ``Moore-Penrose inverse'' of <i>F</i> (see
	171	<a href="biblio.htm#BaCa"><cite>BaCa</cite></a>). Then the <strong>MV-efficiency measure</strong> of <i>D</i> is the minimum
	172	value (over all <i>i</i>,<i>j</i> ∈ {1,…,<i>v</i>}, <i>i</i> ≠ <i>j</i>) of
	173	2/(<i>M</i><sub><i>ii</i></sub>+<i>M</i><sub><i>jj</i></sub>−<i>M</i><sub><i>ij</i></sub>−<i>M</i><sub><i>ji</i></sub>).
	174	<p>
	175	<pre>
	176	gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));;
	177	gap> BlockDesignBlocks(D);
	178	[ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 8, 11 ],
	179	[ 2, 7, 9, 12 ], [ 3, 5, 10, 12 ], [ 3, 6, 9, 11 ], [ 4, 6, 8, 12 ],
	180	[ 4, 7, 10, 11 ] ]
	181	gap> BlockDesignEfficiency(D);
	182	rec( A := 33/41,
	183	CEFpolynomial := x_1^11-9x_1^10+147/4x_1^9-719/8x_1^8+18723/128x_1^7-106\
	184	47/64x_1^6+138159/1024x_1^5-159813/2048x_1^4+2067201/65536x_1^3-556227/655\
	185	36x_1^2+89667/65536x_1-6561/65536, Dpowered := 6561/65536,
	186	Einterval := [ 3/4, 3/4 ] )
	187	gap> BlockDesignEfficiency(D,10^(-4),true);
	188	rec( A := 33/41,
	189	CEFpolynomial := x_1^11-9x_1^10+147/4x_1^9-719/8x_1^8+18723/128x_1^7-106\
	190	47/64x_1^6+138159/1024x_1^5-159813/2048x_1^4+2067201/65536x_1^3-556227/655\
	191	36x_1^2+89667/65536x_1-6561/65536, Dpowered := 6561/65536,
	192	Einterval := [ 3/4, 3/4 ], MV := 3/4 )
	193	</pre>
	194	<p>
	195	<p>
	196	<h2><a name="SECT003">5.3 Computing an interval for a certain real zero of a rational polynomial</a></h2>
	197	<p><p>
	198	We document a DESIGN package utility function used in the calculation
	199	of the <code>Einterval</code> component above, but is more widely applicable.
	200	<p>
	201	<a name = "SSEC003.1"></a>
	202	<li><code>DESIGN_IntervalForLeastRealZero( </code><var>f</var><code>, </code><var>a</var><code>, </code><var>b</var><code>, </code><var>eps</var><code> )</code>
	203	<p>
	204	Suppose that <var>f</var> is a univariate polynomial over the rationals, <var>a</var>,
	205	<var>b</var> are rational numbers with <i>a</i> ≤ <i>b</i> , and <var>eps</var> is a positive
	206	rational number.
	207	<p>
	208	If <var>f</var> has no real zero in the closed interval [<i>a</i> ,<i>b</i> ], then this
	209	function returns the empty list. Otherwise, let α be the least
	210	real zero of <var>f</var> such that <i>a</i> ≤ α ≤ <i>b</i> . Then this function
	211	returns a list [<i>c</i>,<i>d</i>] of rational numbers, with <i>c</i> ≤ α ≤ <i>d</i>
	212	and <i>d</i>−<i>c</i> ≤ <i>eps</i> . Moreover, <i>c</i>=<i>d</i> if and only if α is rational
	213	(in which case α = <i>c</i>=<i>d</i>).
	214	<p>
	215	<pre>
	216	gap> x:=Indeterminate(Rationals,1);
	217	x_1
	218	gap> f:=(x+3)*(x^2-3);
	219	x_1^3+3x_1^2-3x_1-9
	220	gap> L:=DESIGN_IntervalForLeastRealZero(f,-5,5,10^(-3));
	221	[ -3, -3 ]
	222	gap> L:=DESIGN_IntervalForLeastRealZero(f,-2,5,10^(-3));
	223	[ -14193/8192, -7093/4096 ]
	224	gap> List(L,Float);
	225	[ -1.73254, -1.73169 ]
	226	gap> L:=DESIGN_IntervalForLeastRealZero(f,0,5,10^(-3));
	227	[ 14185/8192, 7095/4096 ]
	228	gap> List(L,Float);
	229	[ 1.73157, 1.73218 ]
	230	gap> L:=DESIGN_IntervalForLeastRealZero(f,0,5,10^(-5));
	231	[ 454045/262144, 908095/524288 ]
	232	gap> List(L,Float);
	233	[ 1.73204, 1.73205 ]
	234	gap> L:=DESIGN_IntervalForLeastRealZero(f,2,5,10^(-5));
	235	[ ]
	236	</pre>
	237	<p>
	238	<p>
	239	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href ="CHAP006.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
	240	<P>
	241	<address>design manual<br>March 2019
	242	</address></body></html>⏎

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	0	<html><head><title>[design] 6 Automorphism groups and isomorphism testing for block designs</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP005.htm">Previous</a>] [<a href ="CHAP007.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
	3	<h1>6 Automorphism groups and isomorphism testing for block designs</h1><p>
	4	<P>
	5	<H3>Sections</H3>
	6	<oL>
	7	<li> <A HREF="CHAP006.htm#SECT001">Computing automorphism groups</a>
	8	<li> <A HREF="CHAP006.htm#SECT002">Testing isomorphism</a>
	9	</ol><p>
	10	<p>
	11	The functions in this chapter depend on nauty <a href="biblio.htm#Nau90"><cite>Nau90</cite></a>, <a href="biblio.htm#MP14"><cite>MP14</cite></a>
	12	or bliss <a href="biblio.htm#JK07"><cite>JK07</cite></a> via the GRAPE package, which must be fully
	13	installed in order for these functions to work.
	14	<p>
	15	<p>
	16	<h2><a name="SECT001">6.1 Computing automorphism groups</a></h2>
	17	<p><p>
	18	<a name = "SSEC001.1"></a>
	19	<li><code>AutGroupBlockDesign( </code><var>D</var><code> )</code>
	20	<p>
	21	This function returns the automorphism group of the block design <var>D</var>.
	22	The <strong>automorphism group</strong> \Aut(<i>D</i> ) of <var>D</var> is the group consisting of
	23	all the permutations of the points {1,…,<i>D</i> <tt>.</tt><tt>v</tt>} which preserve
	24	the block-multiset of <var>D</var>.
	25	<p>
	26	This function is not yet implemented for non-binary block designs.
	27	<p>
	28	This function can also be called via <code>AutomorphismGroup(</code><var>D</var><code>)</code>.
	29	<p>
	30	<pre>
	31	gap> D:=PGPointFlatBlockDesign(2,3,1);; # projective plane of order 3
	32	gap> Size(AutGroupBlockDesign(D));
	33	5616
	34	</pre>
	35	<p>
	36	<p>
	37	<h2><a name="SECT002">6.2 Testing isomorphism</a></h2>
	38	<p><p>
	39	<a name = "SSEC002.1"></a>
	40	<li><code>IsIsomorphicBlockDesign( </code><var>D1</var><code>, </code><var>D2</var><code> )</code>
	41	<p>
	42	This boolean function returns <code>true</code> if and only if block designs
	43	<var>D1</var> and <var>D2</var> are <strong>isomorphic</strong>, that is, there is a bijection from the
	44	point-set of <var>D1</var> to that of <var>D2</var> which maps the block-multiset of <var>D1</var>
	45	to that of <var>D2</var>.
	46	<p>
	47	This function is not yet implemented for non-binary block designs.
	48	<p>
	49	For pairwise isomorphism testing for three or more binary block designs,
	50	see <a href="CHAP006.htm#SSEC002.2">BlockDesignIsomorphismClassRepresentatives</a>.
	51	<p>
	52	<pre>
	53	gap> D1:=BlockDesign(3,[[1],[1,2,3],[2]]);;
	54	gap> D2:=BlockDesign(3,[[1],[1,2,3],[3]]);;
	55	gap> IsIsomorphicBlockDesign(D1,D2);
	56	true
	57	gap> D3:=BlockDesign(4,[[1],[1,2,3],[3]]);;
	58	gap> IsIsomorphicBlockDesign(D2,D3);
	59	false
	60	gap> # block designs with different numbers of points are not isomorphic
	61	</pre>
	62	<p>
	63	<a name = "SSEC002.2"></a>
	64	<li><code>BlockDesignIsomorphismClassRepresentatives( </code><var>L</var><code> )</code>
	65	<p>
	66	Given a list <var>L</var> of binary block designs, this function returns a list
	67	consisting of pairwise non-isomorphic elements of <var>L</var>, representing all
	68	the isomorphism classes of elements of <var>L</var>. The order of the elements
	69	in the returned list may differ from their order in <var>L</var>.
	70	<p>
	71	<pre>
	72	gap> D1:=BlockDesign(3,[[1],[1,2,3],[2]]);;
	73	gap> D2:=BlockDesign(3,[[1],[1,2,3],[3]]);;
	74	gap> D3:=BlockDesign(4,[[1],[1,2,3],[3]]);;
	75	gap> BlockDesignIsomorphismClassRepresentatives([D1,D2,D3]);
	76	[ rec( isBlockDesign := true, v := 4, blocks := [ [ 1 ], [ 1, 2, 3 ], [ 3 ] ],
	77	isBinary := true ),
	78	rec( isBlockDesign := true, v := 3, blocks := [ [ 1 ], [ 1, 2, 3 ], [ 2 ] ],
	79	isBinary := true ) ]
	80	</pre>
	81	<p>
	82	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP005.htm">Previous</a>] [<a href ="CHAP007.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
	83	<P>
	84	<address>design manual<br>March 2019
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	3	<h1>7 Classifying block designs</h1><p>
	4	<P>
	5	<H3>Sections</H3>
	6	<oL>
	7	<li> <A HREF="CHAP007.htm#SECT001">The function BlockDesigns</a>
	8	</ol><p>
	9	<p>
	10	This chapter describes the function <code>BlockDesigns</code> which can classify
	11	block designs with given properties. The possible properties a user
	12	can specify are many and varied, and are described below. Depending on
	13	the properties, this function can handle block designs with up to about
	14	20 points (sometimes more and sometimes less, depending on the problem).
	15	<p>
	16	<p>
	17	<h2><a name="SECT001">7.1 The function BlockDesigns</a></h2>
	18	<p><p>
	19	<a name = "SSEC001.1"></a>
	20	<li><code>BlockDesigns( </code><var>param</var><code> )</code>
	21	<p>
	22	This function returns a list <var>DL</var> of block designs whose properties are
	23	specified by the user in the record <var>param</var>. The precise interpretation
	24	of the output depends on <var>param</var>, described below. Only binary designs
	25	are generated by this function, if <code></code><var>param</var><code>.blockDesign</code> is unbound
	26	or is a binary design.
	27	<p>
	28	The required components of <var>param</var> are <code>v</code>, <code>blockSizes</code>, and
	29	<code>tSubsetStructure</code>.
	30	<p>
	31	<code></code><var>param</var><code>.v</code> must be a positive integer, and specifies that for each block
	32	design in the list <var>DL</var>, the points are 1,...,<code></code><var>param</var><code>.v</code>.
	33	<p>
	34	<code></code><var>param</var><code>.blockSizes</code> must be a set of positive integers, and specifies
	35	that the block sizes of each block design in <var>DL</var> will be contained in
	36	<code></code><var>param</var><code>.blockSizes</code>.
	37	<p>
	38	<code></code><var>param</var><code>.tSubsetStructure</code> must be a record, having
	39	components <code>t</code>, <code>partition</code>, and <code>lambdas</code>. Let <var>t</var>
	40	be equal to <code></code><var>param</var><code>.tSubsetStructure.t</code>, <var>partition</var>
	41	be <code></code><var>param</var><code>.tSubsetStructure.partition</code>, and <var>lambdas</var> be
	42	<code></code><var>param</var><code>.tSubsetStructure.lambdas</code>. Then <var>t</var> must be a non-negative
	43	integer, <var>partition</var> must be a list of non-empty sets of <var>t</var>-subsets of
	44	<code>[1..</code><var>param</var><code>.v]</code>, forming an ordered partition of all the <var>t</var>-subsets of
	45	<code>[1..</code><var>param</var><code>.v]</code>, and <var>lambdas</var> must be a list of distinct non-negative
	46	integers (not all zero) of the same length as <var>partition</var>. This specifies
	47	that for each design in <var>DL</var>, each <var>t</var>-subset in <code></code><var>partition</var><code>[</code><var>i</var><code>]</code>
	48	will occur exactly <code></code><var>lambdas</var><code>[</code><var>i</var><code>]</code> times, counted over all blocks of
	49	the design. For binary designs, this means that each <var>t</var>-subset in
	50	<code></code><var>partition</var><code>[</code><var>i</var><code>]</code> is contained in exactly <code></code><var>lambdas</var><code>[</code><var>i</var><code>]</code> blocks.
	51	The <code>partition</code> component is optional if <var>lambdas</var> has length 1.
	52	We require that <var>t</var> is less than or equal to each element of
	53	<code></code><var>param</var><code>.blockSizes</code>, and if <code></code><var>param</var><code>.blockDesign</code> is bound,
	54	then each block of <code></code><var>param</var><code>.blockDesign</code> must contain at least <var>t</var>
	55	distinct elements. Note that if <code></code><var>param</var><code>.tSubsetStructure</code> is equal to
	56	<code>rec(t:=0,lambdas:=[</code><var>b</var><code>])</code>, for some positive integer <var>b</var>, then all
	57	that is being specified is that each design in <var>DL</var> must have exactly
	58	<var>b</var> blocks.
	59	<p>
	60	The optional components of <var>param</var> are used to specify additional
	61	constraints on the designs in <var>DL</var> or to change default parameter
	62	values. These optional components are <code>blockDesign</code>, <code>r</code>, <code>b</code>,
	63	<code>blockNumbers</code>, <code>blockIntersectionNumbers</code>, <code>blockMaxMultiplicities</code>,
	64	<code>isoGroup</code>, <code>requiredAutSubgroup</code>, and <code>isoLevel</code>.
	65	<p>
	66	<code></code><var>param</var><code>.blockDesign</code> must be a block design with <code></code><var>param</var><code>.blockDesign.v</code>
	67	equal to <code></code><var>param</var><code>.v</code>. Then each block multiset of a design in <var>DL</var> will
	68	be a submultiset of <code></code><var>param</var><code>.blockDesign.blocks</code> (that is, each block of
	69	a design <var>D</var> in <var>DL</var> will be a block of <code></code><var>param</var><code>.blockDesign</code>, and the
	70	multiplicity of a block of <var>D</var> will be less than or equal to that block's
	71	multiplicity in <code></code><var>param</var><code>.blockDesign</code>). The <code>blockDesign</code> component is
	72	useful for the computation of subdesigns, such as parallel classes.
	73	<p>
	74	<code></code><var>param</var><code>.r</code> must be a positive integer, and specifies that in each design
	75	in <var>DL</var>, each point will occur exactly <code></code><var>param</var><code>.r</code> times in the list of
	76	blocks. In other words, each design in <var>DL</var> will have replication number
	77	<code></code><var>param</var><code>.r</code>.
	78	<p>
	79	<code></code><var>param</var><code>.b</code> must be a positive integer, and specifies that each design
	80	in <var>DL</var> will have exactly <code></code><var>param</var><code>.b</code> blocks.
	81	<p>
	82	<code></code><var>param</var><code>.blockNumbers</code> must be a list of non-negative integers, the <var>i</var>-th
	83	element of which specifies the number of blocks whose size is equal
	84	to <code></code><var>param</var><code>.blockSizes[</code><var>i</var><code>]</code> (for each design in <var>DL</var>). The length of
	85	<code></code><var>param</var><code>.blockNumbers</code> must equal that of <code></code><var>param</var><code>.blockSizes</code>, and at
	86	least one entry of <code></code><var>param</var><code>.blockNumbers</code> must be positive.
	87	<p>
	88	<code></code><var>param</var><code>.blockIntersectionNumbers</code> must be a symmetric matrix of sets
	89	of non-negative integers, the <code>[</code><var>i</var><code>][</code><var>j</var><code>]</code>-element of which specifies
	90	the set of possible sizes for the intersection of a block <i>B</i> of
	91	size <code></code><var>param</var><code>.blockSizes[</code><var>i</var><code>]</code> with a different block (but possibly
	92	a repeat of <i>B</i>) of size <code></code><var>param</var><code>.blockSizes[</code><var>j</var><code>]</code> (for each design
	93	in <var>DL</var>). In the case of multisets, we take the multiplicity of an
	94	element in the intersection to be the minimum of its multiplicities in
	95	the multisets being intersected; for example, the intersection of
	96	<code>[1,1,1,2,2,3]</code> with <code>[1,1,2,2,2,4]</code> is <code>[1,1,2,2]</code>, having size 4. The
	97	dimension of <code></code><var>param</var><code>.blockIntersectionNumbers</code> must equal the length of
	98	<code></code><var>param</var><code>.blockSizes</code>.
	99	<p>
	100	<code></code><var>param</var><code>.blockMaxMultiplicities</code> must be a list of non-negative integers,
	101	the <var>i</var>-th element of which specifies an upper bound on the multiplicity
	102	of a block whose size is equal to <code></code><var>param</var><code>.blockSizes[</code><var>i</var><code>]</code> (for each
	103	design in <var>DL</var>). The length of <code></code><var>param</var><code>.blockMaxMultiplicities</code> must
	104	equal that of <code></code><var>param</var><code>.blockSizes</code>.
	105	<p>
	106	Let <var>G</var> be the automorphism group of <code></code><var>param</var><code>.blockDesign</code> if bound, and
	107	<var>G</var> be <code>SymmetricGroup(</code><var>param</var><code>.v)</code> otherwise. Let <var>H</var> be the subgroup of
	108	<var>G</var> stabilizing <code></code><var>param</var><code>.tSubsetStructure.partition</code> (as an ordered list
	109	of sets of sets) if bound, and <var>H</var> be equal to <var>G</var> otherwise.
	110	<p>
	111	<code></code><var>param</var><code>.isoGroup</code> must be a subgroup of <var>H</var>, and specifies that we
	112	consider two designs with the required properties to be <strong>equivalent</strong>
	113	if their block multisets are in the same orbit of <code></code><var>param</var><code>.isoGroup</code>
	114	(in its action on multisets of multisets of <code>[1..</code><var>param</var><code>.v]</code>). The
	115	default for <code></code><var>param</var><code>.isoGroup</code> is <var>H</var>. Thus, if <code></code><var>param</var><code>.blockDesign</code>
	116	and <code></code><var>param</var><code>.isoGroup</code> are both unbound, equivalence is the same as
	117	block-design isomorphism for the required designs.
	118	<p>
	119	<code></code><var>param</var><code>.requiredAutSubgroup</code> must be a subgroup of <code></code><var>param</var><code>.isoGroup</code>,
	120	and specifies that each design in <var>DL</var> must be invariant under
	121	<code></code><var>param</var><code>.requiredAutSubgroup</code> (in its action on multisets of multisets of
	122	<code>[1..</code><var>param</var><code>.v]</code>). The default for <code></code><var>param</var><code>.requiredAutSubgroup</code> is the
	123	trivial permutation group.
	124	<p>
	125	<code></code><var>param</var><code>.isoLevel</code> must be 0, 1, or 2 (the default is 2). The value
	126	0 specifies that <var>DL</var> will contain at most one block design, and will
	127	contain one block design with the required properties if and only if
	128	such a block design exists; the value 1 specifies that <var>DL</var> will contain
	129	(perhaps properly) a list of <code></code><var>param</var><code>.isoGroup</code>-orbit representatives of
	130	the required designs; the value 2 specifies that <var>DL</var> will consist
	131	precisely of <code></code><var>param</var><code>.isoGroup</code>-orbit representatives of the required
	132	designs.
	133	<p>
	134	For an example, we classify up to isomorphism the 2-(15,3,1) designs
	135	invariant under a semi-regular group of automorphisms of order 5, and
	136	then classify all parallel classes of these designs, up to the action
	137	of the automorphism groups of these designs.
	138	<p>
	139	<pre>
	140	gap> DL:=BlockDesigns(rec(
	141	> v:=15,blockSizes:=[3],
	142	> tSubsetStructure:=rec(t:=2,lambdas:=[1]),
	143	> requiredAutSubgroup:=
	144	> Group((1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15))));;
	145	gap> List(DL,AllTDesignLambdas);
	146	[ [ 35, 7, 1 ], [ 35, 7, 1 ], [ 35, 7, 1 ] ]
	147	gap> List(DL,D->Size(AutGroupBlockDesign(D)));
	148	[ 20160, 5, 60 ]
	149	gap> parclasses:=List(DL,D->
	150	> BlockDesigns(rec(
	151	> blockDesign:=D,
	152	> v:=15,blockSizes:=[3],
	153	> tSubsetStructure:=rec(t:=1,lambdas:=[1]))));
	154	[ [ rec( isBlockDesign := true, v := 15,
	155	blocks := [ [ 1, 2, 6 ], [ 3, 4, 8 ], [ 5, 7, 14 ], [ 9, 12, 15 ],
	156	[ 10, 11, 13 ] ],
	157	tSubsetStructure := rec( t := 1, lambdas := [ 1 ] ),
	158	isBinary := true, isSimple := true, blockSizes := [ 3 ],
	159	blockNumbers := [ 5 ], r := 1,
	160	autSubgroup := Group([ (2,6)(3,11)(4,10)(5,14)(8,13)(12,15),
	161	(2,6)(4,8)(5,12)(7,9)(10,13)(14,15),
	162	(2,6)(3,12)(4,9)(7,14)(8,15)(11,13),
	163	(3,12,5)(4,15,7)(8,9,14)(10,11,13),
	164	(1,6,2)(3,4,8)(5,7,14)(9,12,15)(10,11,13),
	165	(1,8,11,2,3,10)(4,13,6)(5,15,14,9,7,12) ]) ) ],
	166	[ rec( isBlockDesign := true, v := 15,
	167	blocks := [ [ 1, 7, 12 ], [ 2, 8, 13 ], [ 3, 9, 14 ],
	168	[ 4, 10, 15 ], [ 5, 6, 11 ] ],
	169	tSubsetStructure := rec( t := 1, lambdas := [ 1 ] ),
	170	isBinary := true, isSimple := true, blockSizes := [ 3 ],
	171	blockNumbers := [ 5 ], r := 1,
	172	autSubgroup := Group([ (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,12) ]) )
	173	],
	174	[ rec( isBlockDesign := true, v := 15, blocks := [ [ 1, 2, 6 ], [ 3, 10, 13
	175	], [ 4, 11, 12 ], [ 5, 7, 15 ], [ 8, 9, 14 ] ],
	176	tSubsetStructure := rec( t := 1, lambdas := [ 1 ] ),
	177	isBinary := true, isSimple := true, blockSizes := [ 3 ],
	178	blockNumbers := [ 5 ], r := 1,
	179	autSubgroup := Group([ (1,2)(3,5)(7,10)(8,9)(11,12)(13,15),
	180	(1,11,8)(2,12,9)(3,13,10)(4,14,6)(5,15,7) ]) ),
	181	rec( isBlockDesign := true, v := 15,
	182	blocks := [ [ 1, 8, 11 ], [ 2, 9, 12 ], [ 3, 10, 13 ],
	183	[ 4, 6, 14 ], [ 5, 7, 15 ] ],
	184	tSubsetStructure := rec( t := 1, lambdas := [ 1 ] ),
	185	isBinary := true, isSimple := true, blockSizes := [ 3 ],
	186	blockNumbers := [ 5 ], r := 1,
	187	autSubgroup := Group([ (1,2)(3,5)(7,10)(8,9)(11,12)(13,15),
	188	(1,3,4,2)(6,9,8,10)(11,13,14,12),
	189	(1,3,5,2,4)(6,8,10,7,9)(11,13,15,12,14),
	190	(1,11,8)(2,12,9)(3,13,10)(4,14,6)(5,15,7) ]) ) ] ]
	191	gap> List(parclasses,Length);
	192	[ 1, 1, 2 ]
	193	gap> List(parclasses,L->List(L,parclass->Size(parclass.autSubgroup)));
	194	[ [ 360 ], [ 5 ], [ 6, 60 ] ]
	195	</pre>
	196	<p>
	197	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP006.htm">Previous</a>] [<a href ="CHAP008.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
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	0	<html><head><title>[design] 8 Classifying semi-Latin squares</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
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	3	<h1>8 Classifying semi-Latin squares</h1><p>
	4	<P>
	5	<H3>Sections</H3>
	6	<oL>
	7	<li> <A HREF="CHAP008.htm#SECT001">Semi-Latin squares and SOMAs</a>
	8	<li> <A HREF="CHAP008.htm#SECT002">The function SemiLatinSquareDuals</a>
	9	</ol><p>
	10	<p>
	11	This chapter describes the function <code>SemiLatinSquareDuals</code> which can
	12	classify semi-Latin squares with certain given properties, and return
	13	a list of their duals as block designs.
	14	<p>
	15	<p>
	16	<h2><a name="SECT001">8.1 Semi-Latin squares and SOMAs</a></h2>
	17	<p><p>
	18	Let <i>n</i> and <i>k</i> be positive integers. An (<i>n</i>×<i>n</i>)/<i>k</i> <strong>semi-Latin
	19	square</strong>
	20	<a name = "I0"></a>
	21
	22	is an <i>n</i> by <i>n</i> array <i>A</i>, whose
	23	entries are <i>k</i>-subsets of a <i>kn</i>-set <i>X</i> (the <strong>symbol-set</strong>), such that
	24	each element of <i>X</i> occurs exactly once in each row and exactly once in
	25	each column of <i>A</i>. (Thus an (<i>n</i>×<i>n</i>)/1 semi-Latin square is the same
	26	thing as a Latin square of order <i>n</i>.) For extensive useful information on
	27	semi-Latin squares, see <a href="http://www.maths.qmul.ac.uk/~rab/sls.html">http://www.maths.qmul.ac.uk/~rab/sls.html</a>.
	28	<p>
	29	A SOMA(<i>k</i>,<i>n</i>)
	30	<a name = "I1"></a>
	31
	32	is an (<i>n</i>×<i>n</i>)/<i>k</i> semi-Latin square <i>A</i>,
	33	with <i>n</i> ≥ 2, in which no 2-subset of the symbol-set is contained in
	34	more than one entry of <i>A</i>. For extensive useful information on SOMAs,
	35	see <a href="http://www.maths.qmul.ac.uk/~lsoicher/soma/">http://www.maths.qmul.ac.uk/~lsoicher/soma/</a>.
	36	<p>
	37	Let <i>A</i> and <i>B</i> be (<i>n</i>×<i>n</i>)/<i>k</i> semi-Latin squares. We say that
	38	<i>B</i> is <strong>(weakly) isomorphic</strong> to <i>A</i> if <i>B</i> can be obtained from <i>A</i>
	39	by applying one or more of: a row permutation; a column permutation;
	40	transposing; renaming the symbols. If transposing is not allowed then we
	41	get the concept of strong isomorphism. More formally, <i>B</i> is <strong>strongly
	42	isomorphic</strong> to <i>A</i> if <i>B</i> can be obtained from <i>A</i> by applying one or
	43	more of: a row permutation; a column permutation; renaming the symbols.
	44	<p>
	45	Let <i>A</i> be an (<i>n</i>×<i>n</i>)/<i>k</i> semi-Latin square. Then the dual of <i>A</i>
	46	can be represented as a binary block design as follows. The point-set of
	47	<i>D</i> is taken to be the Cartesian square of {1,…,<i>n</i>}, with [<i>x</i>,<i>y</i>]
	48	representing the [<i>x</i>,<i>y</i>]-entry of <i>A</i>. The blocks of <i>D</i> are in one-to-one
	49	correspondance with the symbols of <i>A</i>, with the <i>i</i>-th block of <i>D</i>
	50	consisting of the ordered pairs [<i>x</i>,<i>y</i>] such that the <i>i</i>-th symbol of
	51	<i>A</i> is contained in the [<i>x</i>,<i>y</i>]-entry of <i>A</i>. Given <i>D</i>, the semi-Latin
	52	square <i>A</i> can be recovered, up to the naming of its symbols.
	53	<p>
	54	<p>
	55	<h2><a name="SECT002">8.2 The function SemiLatinSquareDuals</a></h2>
	56	<p><p>
	57	<a name = "SSEC002.1"></a>
	58	<li><code>SemiLatinSquareDuals( </code><var>n</var><code>, </code><var>k</var><code> )</code>
	59	<li><code>SemiLatinSquareDuals( </code><var>n</var><code>, </code><var>k</var><code>, </code><var>maxmult</var><code> )</code>
	60	<li><code>SemiLatinSquareDuals( </code><var>n</var><code>, </code><var>k</var><code>, </code><var>maxmult</var><code>, </code><var>blockintsizes</var><code> )</code>
	61	<li><code>SemiLatinSquareDuals( </code><var>n</var><code>, </code><var>k</var><code>, </code><var>maxmult</var><code>, </code><var>blockintsizes</var><code>, </code><var>isolevel</var><code> )</code>
	62	<p>
	63	Let <var>n</var> and <var>k</var> be positive integers. Then this function (which makes
	64	heavy use of the function <code>BlockDesigns</code>) returns a list <var>DL</var> of block
	65	designs which are the duals of the (<i>n</i> ×<i>n</i> )/<i>k</i> semi-Latin
	66	squares whose properties are specified by the given parameters, described
	67	below. In practice, depending on the specified properties, this function
	68	can be useful for <var>n</var> up to about 6 or 7.
	69	<p>
	70	The parameter <var>maxmult</var>, if given, must be a positive integer or
	71	the string <code>"default"</code>. If it is a positive integer, then <var>maxmult</var>
	72	specifies an upper bound on the multiplicity of each block in each
	73	semi-Latin square dual in <var>DL</var>. The default value for <var>maxmult</var> (if
	74	omitted or if given as <code>"default"</code>) is <var>k</var>, which poses no constraint
	75	on the block multiplicities.
	76	<p>
	77	The parameter <var>blockintsizes</var>, if given, must be a set of non-negative
	78	integers or the string <code>"default"</code>. If it is given as a set, then
	79	<var>blockintsizes</var> specifies, for each semi-Latin square dual in <var>DL</var>,
	80	the set of possible sizes for the intersection of a block <i>B</i> with a
	81	different block (but possibly a repeat of <i>B</i>). The default value for
	82	<var>blockintsizes</var> (if omitted or if given as <code>"default"</code>) is <code>[0..</code><var>n</var><code>]</code>,
	83	which poses no constraint on the block intersection sizes. Note that
	84	block intersection sizes in the dual of a semi-Latin square correspond
	85	to concurrencies of points in the semi-Latin square itself. Also note
	86	that if <i>n</i> ≥ 2 and <var>blockintsizes</var> is specified to be <code>[0,1]</code> then
	87	the (<i>n</i>×<i>n</i>)/<i>k</i> semi-Latin squares being considered are SOMA(<i>k</i>,<i>n</i>)s.
	88	<p>
	89	The parameter <var>isolevel</var>, if given, must be 0, 1, 2, 3, 4 or the string
	90	<code>"default"</code> (the default value is 2). The value 0 specifies that <var>DL</var>
	91	will contain at most one (semi-Latin square dual given as a) block design,
	92	and will contain one such block design if and only if a semi-Latin square
	93	with the required properties exists. The value 1 specifies that <var>DL</var>
	94	will contain a list of duals representing all weak isomorphism classes
	95	of semi-Latin squares with the required properties (possibly with some
	96	classes represented more than once) and the value 2 specifies that <var>DL</var>
	97	will contain precisely one dual semi-Latin square representative for
	98	each weak isomorphism class of semi-Latin squares with the required
	99	properties. The values 3 and 4 for <var>isolevel</var> play the roles of 1 and 2,
	100	respectively, but with weak isomorphism replaced by strong isomorphism.
	101	Thus, <i>isolevel</i> =3 specifies that <var>DL</var> will contain a list of duals
	102	representing all strong isomorphism classes of semi-Latin squares with
	103	the required properties (possibly with some classes represented more than
	104	once) and <i>isolevel</i> =4 specifies that <var>DL</var> will contain precisely one
	105	dual semi-Latin square representative for each strong isomorphism class
	106	of semi-Latin squares with the required properties.
	107	<p>
	108	For example, we determine the numbers of weak and strong isomorphism
	109	classes of (4×4)/<i>k</i> semi-Latin squares for <i>k</i>=1,…,6. (These
	110	numbers disagree with P. E. Chigbu's classification for the cases <i>k</i>=3,4
	111	<a href="biblio.htm#BaCh"><cite>BaCh</cite></a>.)
	112	<p>
	113	<pre>
	114	gap> List([1..6],k->Length(SemiLatinSquareDuals(4,k))); # weak
	115	[ 2, 10, 40, 164, 621, 2298 ]
	116	gap> List([1..6],k->Length(SemiLatinSquareDuals(4,k,"default","default",4))); # strong
	117	[ 2, 11, 46, 201, 829, 3343 ]
	118	</pre>
	119	<p>
	120	Next, we determine one SOMA(3,6).
	121	<p>
	122	<pre>
	123	gap> SemiLatinSquareDuals(6,3,"default",[0,1],0);
	124	[ rec( isBlockDesign := true, v := 36,
	125	blocks := [ [ 1, 8, 15, 22, 29, 36 ], [ 1, 9, 16, 23, 30, 32 ],
	126	[ 1, 12, 14, 21, 28, 35 ], [ 2, 9, 17, 24, 25, 34 ],
	127	[ 2, 11, 18, 22, 27, 31 ], [ 2, 12, 16, 19, 29, 33 ],
	128	[ 3, 10, 14, 24, 29, 31 ], [ 3, 11, 16, 20, 25, 36 ],
	129	[ 3, 12, 13, 23, 26, 34 ], [ 4, 7, 14, 23, 27, 36 ],
	130	[ 4, 8, 17, 21, 30, 31 ], [ 4, 9, 18, 19, 26, 35 ],
	131	[ 5, 7, 15, 20, 30, 34 ], [ 5, 8, 13, 24, 28, 33 ],
	132	[ 5, 10, 18, 21, 25, 32 ], [ 6, 7, 17, 22, 26, 33 ],
	133	[ 6, 10, 13, 20, 27, 35 ], [ 6, 11, 15, 19, 28, 32 ] ],
	134	tSubsetStructure := rec( t := 1, lambdas := [ 3 ] ), isBinary := true,
	135	isSimple := true, blockSizes := [ 6 ], blockNumbers := [ 18 ], r := 3,
	136	autSubgroup := <permutation group of size 72 with 3 generators>,
	137	pointNames := [ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ],
	138	[ 1, 6 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 2, 4 ], [ 2, 5 ],
	139	[ 2, 6 ], [ 3, 1 ], [ 3, 2 ], [ 3, 3 ], [ 3, 4 ], [ 3, 5 ],
	140	[ 3, 6 ], [ 4, 1 ], [ 4, 2 ], [ 4, 3 ], [ 4, 4 ], [ 4, 5 ],
	141	[ 4, 6 ], [ 5, 1 ], [ 5, 2 ], [ 5, 3 ], [ 5, 4 ], [ 5, 5 ],
	142	[ 5, 6 ], [ 6, 1 ], [ 6, 2 ], [ 6, 3 ], [ 6, 4 ], [ 6, 5 ],
	143	[ 6, 6 ] ] ) ]
	144	</pre>
	145	<p>
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	148	<address>design manual<br>March 2019
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	0	<html><head><title>[design] 9 Partitioning block designs</title></head>
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	3	<h1>9 Partitioning block designs</h1><p>
	4	<P>
	5	<H3>Sections</H3>
	6	<oL>
	7	<li> <A HREF="CHAP009.htm#SECT001">Partitioning a block design into block designs</a>
	8	<li> <A HREF="CHAP009.htm#SECT002">Computing resolutions</a>
	9	</ol><p>
	10	<p>
	11	This chapter describes the function <code>PartitionsIntoBlockDesigns</code> which can
	12	classify partitions of (the block multiset of) a given block design into
	13	(the block multisets of) block designs having user-specified properties.
	14	We also describe <code>MakeResolutionsComponent</code> which is useful for the
	15	special case when the desired partitions are resolutions.
	16	<p>
	17	<p>
	18	<h2><a name="SECT001">9.1 Partitioning a block design into block designs</a></h2>
	19	<p><p>
	20	<a name = "SSEC001.1"></a>
	21	<li><code>PartitionsIntoBlockDesigns( </code><var>param</var><code> )</code>
	22	<p>
	23	Let <var>D</var> equal <code></code><var>param</var><code>.blockDesign</code>. This function returns a list <var>PL</var>
	24	of partitions of (the block multiset of) <var>D</var>. Each element of <var>PL</var> is a
	25	record with one component <code>partition</code>, and, in most cases, a component
	26	<code>autGroup</code>. The <code>partition</code> component gives a list <var>P</var> of block designs,
	27	all with the same point set as <var>D</var>, such that the list of (the block
	28	multisets of) the designs in <code></code><var>P</var><code>.partition</code> forms a partition of (the
	29	block multiset of) <var>D</var>. The component <code></code><var>P</var><code>.autGroup</code>, if bound, gives
	30	the automorphism group of the partition, which is the stabilizer of the
	31	partition in the automorphism group of <var>D</var>. The precise interpretation
	32	of the output depends on <var>param</var>, described below.
	33	<p>
	34	The required components of <var>param</var> are <code>blockDesign</code>, <code>v</code>, <code>blockSizes</code>,
	35	and <code>tSubsetStructure</code>.
	36	<p>
	37	<code></code><var>param</var><code>.blockDesign</code> is the block design to be partitioned.
	38	<p>
	39	<code></code><var>param</var><code>.v</code> must be a positive integer, and specifies that for each block
	40	design in each partition in <var>PL</var>, the points are 1,...,<code></code><var>param</var><code>.v</code>.
	41	It is required that <code></code><var>param</var><code>.v</code> be equal to <code></code><var>param</var><code>.blockDesign.v</code>.
	42	<p>
	43	<code></code><var>param</var><code>.blockSizes</code> must be a set of positive integers, and specifies
	44	that the block sizes of each block design in each partition in <var>PL</var>
	45	will be contained in <code></code><var>param</var><code>.blockSizes</code>.
	46	<p>
	47	<code></code><var>param</var><code>.tSubsetStructure</code> must be a record, having
	48	components <code>t</code>, <code>partition</code>, and <code>lambdas</code>. Let <var>t</var> be
	49	equal to <code></code><var>param</var><code>.tSubsetStructure.t</code>, <var>partition</var> be
	50	<code></code><var>param</var><code>.tSubsetStructure.partition</code>, and <var>lambdas</var> be
	51	<code></code><var>param</var><code>.tSubsetStructure.lambdas</code>. Then <var>t</var> must be a non-negative
	52	integer, <var>partition</var> must be a list of non-empty sets of <var>t</var>-subsets of
	53	<code>[1..</code><var>param</var><code>.v]</code>, forming an ordered partition of all the <var>t</var>-subsets of
	54	<code>[1..</code><var>param</var><code>.v]</code>, and <var>lambdas</var> must be a list of distinct non-negative
	55	integers (not all zero) of the same length as <var>partition</var>. This specifies
	56	that for each design in each partition in <var>PL</var>, each <var>t</var>-subset in
	57	<code></code><var>partition</var><code>[</code><var>i</var><code>]</code> will occur exactly <code></code><var>lambdas</var><code>[</code><var>i</var><code>]</code> times, counted
	58	over all blocks of the design. For binary designs, this means that each
	59	<var>t</var>-subset in <code></code><var>partition</var><code>[</code><var>i</var><code>]</code> is contained in exactly <code></code><var>lambdas</var><code>[</code><var>i</var><code>]</code>
	60	blocks. The <code>partition</code> component is optional if <var>lambdas</var> has length 1.
	61	We require that <var>t</var> is less than or equal to each element of
	62	<code></code><var>param</var><code>.blockSizes</code>, and that each block of <code></code><var>param</var><code>.blockDesign</code>
	63	contains at least <var>t</var> distinct elements.
	64	<p>
	65	The optional components of <var>param</var> are used to specify additional
	66	constraints on the partitions in <var>PL</var>, or to change default parameter
	67	values. These optional components are <code>r</code>, <code>b</code>, <code>blockNumbers</code>,
	68	<code>blockIntersectionNumbers</code>, <code>blockMaxMultiplicities</code>, <code>isoGroup</code>,
	69	<code>requiredAutSubgroup</code>, and <code>isoLevel</code>. Note that the last three of these
	70	optional components refer to the partitions and not to the block designs
	71	in a partition.
	72	<p>
	73	<code></code><var>param</var><code>.r</code> must be a positive integer, and specifies that in each design
	74	in each partition in <var>PL</var>, each point must occur exactly <code></code><var>param</var><code>.r</code>
	75	times in the list of blocks.
	76	<p>
	77	<code></code><var>param</var><code>.b</code> must be a positive integer, and specifies that each design
	78	in each partition in <var>PL</var> has exactly <code></code><var>param</var><code>.b</code> blocks.
	79	<p>
	80	<code></code><var>param</var><code>.blockNumbers</code> must be a list of non-negative integers, the <var>i</var>-th
	81	element of which specifies the number of blocks whose size is equal to
	82	<code></code><var>param</var><code>.blockSizes[</code><var>i</var><code>]</code> (in each design in each partition in <var>PL</var>). The
	83	length of <code></code><var>param</var><code>.blockNumbers</code> must equal that of <code></code><var>param</var><code>.blockSizes</code>,
	84	and at least one entry of <code></code><var>param</var><code>.blockNumbers</code> must be positive.
	85	<p>
	86	<code></code><var>param</var><code>.blockIntersectionNumbers</code> must be a symmetric matrix of sets
	87	of non-negative integers, the <code>[</code><var>i</var><code>][</code><var>j</var><code>]</code>-element of which specifies
	88	the set of possible sizes for the intersection of a block <i>B</i> of size
	89	<code></code><var>param</var><code>.blockSizes[</code><var>i</var><code>]</code> with a different block (but possibly a repeat of
	90	<i>B</i>) of size <code></code><var>param</var><code>.blockSizes[</code><var>j</var><code>]</code> (in each design in each partition
	91	in <var>PL</var>). In the case of multisets, we take the multiplicity of an
	92	element in the intersection to be the minimum of its multiplicities in
	93	the multisets being intersected; for example, the intersection of
	94	<code>[1,1,1,2,2,3]</code> with <code>[1,1,2,2,2,4]</code> is <code>[1,1,2,2]</code>, having size 4.
	95	The dimension of <code></code><var>param</var><code>.blockIntersectionNumbers</code> must equal the length
	96	of <code></code><var>param</var><code>.blockSizes</code>.
	97	<p>
	98	<code></code><var>param</var><code>.blockMaxMultiplicities</code> must be a list of non-negative
	99	integers, the <var>i</var>-th element of which specifies an upper bound on the
	100	multiplicity of a block whose size is equal to <code></code><var>param</var><code>.blockSizes[</code><var>i</var><code>]</code>
	101	(for each design in each partition in <var>PL</var>). The length of
	102	<code></code><var>param</var><code>.blockMaxMultiplicities</code> must equal that of <code></code><var>param</var><code>.blockSizes</code>.
	103	<p>
	104	<code></code><var>param</var><code>.isoGroup</code> must be a subgroup of the automorphism group of
	105	<code></code><var>param</var><code>.blockDesign</code>. We consider two elements of <var>PL</var> to be
	106	<strong>equivalent</strong> if they are in the same orbit of <code></code><var>param</var><code>.isoGroup</code>
	107	(in its action on multisets of block multisets). The default for
	108	<code></code><var>param</var><code>.isoGroup</code> is the automorphism group of <code></code><var>param</var><code>.blockDesign</code>.
	109	<p>
	110	<code></code><var>param</var><code>.requiredAutSubgroup</code> must be a subgroup of <code></code><var>param</var><code>.isoGroup</code>,
	111	and specifies that each partition in <var>PL</var> must be invariant under
	112	<code></code><var>param</var><code>.requiredAutSubgroup</code> (in its action on multisets of block
	113	multisets). The default for <code></code><var>param</var><code>.requiredAutSubgroup</code> is the trivial
	114	permutation group.
	115	<p>
	116	<code></code><var>param</var><code>.isoLevel</code> must be 0, 1, or 2 (the default is 2). The value 0
	117	specifies that <var>PL</var> will contain at most one partition, and will contain
	118	one partition with the required properties if and only if such a partition
	119	exists; the value 1 specifies that <var>PL</var> will contain (perhaps properly)
	120	a list of <code></code><var>param</var><code>.isoGroup</code> orbit-representatives of the required
	121	partitions; the value 2 specifies that <var>PL</var> will consist precisely of
	122	<code></code><var>param</var><code>.isoGroup</code>-orbit representatives of the required partitions.
	123	<p>
	124	For an example, we first classify up to isomorphism the 2-(15,3,1)
	125	designs invariant under a semi-regular group of automorphisms of order 5,
	126	and then use <code>PartitionsIntoBlockDesigns</code> to classify all the resolutions
	127	of these designs, up to the actions of the respective automorphism groups
	128	of the designs.
	129	<p>
	130	<pre>
	131	gap> DL:=BlockDesigns(rec(
	132	> v:=15,blockSizes:=[3],
	133	> tSubsetStructure:=rec(t:=2,lambdas:=[1]),
	134	> requiredAutSubgroup:=
	135	> Group((1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15))));;
	136	gap> List(DL,D->Size(AutGroupBlockDesign(D)));
	137	[ 20160, 5, 60 ]
	138	gap> PL:=PartitionsIntoBlockDesigns(rec(
	139	> blockDesign:=DL[1],
	140	> v:=15,blockSizes:=[3],
	141	> tSubsetStructure:=rec(t:=1,lambdas:=[1])));
	142	[ rec(
	143	partition := [ rec( isBlockDesign := true, v := 15, blocks := [ [ 1, 2,
	144	6 ], [ 3, 4, 8 ], [ 5, 7, 14 ], [ 9, 12, 15 ],
	145	[ 10, 11, 13 ] ] ),
	146	rec( isBlockDesign := true, v := 15, blocks :=
	147	[ [ 1, 3, 11 ], [ 2, 4, 12 ], [ 5, 6, 8 ], [ 7, 13, 15 ],
	148	[ 9, 10, 14 ] ] ),
	149	rec( isBlockDesign := true, v := 15, blocks :=
	150	[ [ 1, 4, 14 ], [ 2, 5, 15 ], [ 3, 10, 12 ], [ 6, 7, 11 ],
	151	[ 8, 9, 13 ] ] ),
	152	rec( isBlockDesign := true, v := 15, blocks :=
	153	[ [ 1, 5, 10 ], [ 2, 9, 11 ], [ 3, 14, 15 ], [ 4, 6, 13 ],
	154	[ 7, 8, 12 ] ] ),
	155	rec( isBlockDesign := true, v := 15, blocks :=
	156	[ [ 1, 7, 9 ], [ 2, 8, 10 ], [ 3, 5, 13 ], [ 4, 11, 15 ],
	157	[ 6, 12, 14 ] ] ),
	158	rec( isBlockDesign := true, v := 15, blocks :=
	159	[ [ 1, 8, 15 ], [ 2, 13, 14 ], [ 3, 6, 9 ], [ 4, 7, 10 ],
	160	[ 5, 11, 12 ] ] ),
	161	rec( isBlockDesign := true, v := 15, blocks :=
	162	[ [ 1, 12, 13 ], [ 2, 3, 7 ], [ 4, 5, 9 ], [ 6, 10, 15 ],
	163	[ 8, 11, 14 ] ] ) ],
	164	autGroup := Group([ (1,10)(2,11)(3,8)(6,13)(7,14)(12,15),
	165	(1,13)(2,11)(3,14)(4,5)(6,10)(7,8),
	166	(1,13,7)(2,11,5)(6,10,14)(9,12,15),
	167	(2,11,5,15,4,9,12)(3,10,8,14,7,13,6) ]) ),
	168	rec( partition := [ rec( isBlockDesign := true, v := 15,
	169	blocks := [ [ 1, 2, 6 ], [ 3, 4, 8 ], [ 5, 7, 14 ],
	170	[ 9, 12, 15 ], [ 10, 11, 13 ] ] ),
	171	rec( isBlockDesign := true, v := 15,
	172	blocks := [ [ 1, 3, 11 ], [ 2, 4, 12 ], [ 5, 6, 8 ],
	173	[ 7, 13, 15 ], [ 9, 10, 14 ] ] ),
	174	rec( isBlockDesign := true, v := 15,
	175	blocks := [ [ 1, 4, 14 ], [ 2, 5, 15 ], [ 3, 10, 12 ],
	176	[ 6, 7, 11 ], [ 8, 9, 13 ] ] ),
	177	rec( isBlockDesign := true, v := 15,
	178	blocks := [ [ 1, 5, 10 ], [ 2, 13, 14 ], [ 3, 6, 9 ],
	179	[ 4, 11, 15 ], [ 7, 8, 12 ] ] ),
	180	rec( isBlockDesign := true, v := 15,
	181	blocks := [ [ 1, 7, 9 ], [ 2, 8, 10 ], [ 3, 14, 15 ],
	182	[ 4, 6, 13 ], [ 5, 11, 12 ] ] ),
	183	rec( isBlockDesign := true, v := 15,
	184	blocks := [ [ 1, 8, 15 ], [ 2, 9, 11 ], [ 3, 5, 13 ],
	185	[ 4, 7, 10 ], [ 6, 12, 14 ] ] ),
	186	rec( isBlockDesign := true, v := 15,
	187	blocks := [ [ 1, 12, 13 ], [ 2, 3, 7 ], [ 4, 5, 9 ],
	188	[ 6, 10, 15 ], [ 8, 11, 14 ] ] ) ],
	189	autGroup := Group([ (1,15)(2,9)(3,4)(5,7)(6,12)(10,13),
	190	(1,12)(2,9)(3,5)(4,7)(6,15)(8,14),
	191	(1,14)(2,5)(3,8)(6,7)(9,12)(10,13),
	192	(1,8,10)(2,5,15)(3,14,13)(4,9,12) ]) ) ]
	193	gap> List(PL,resolution->Size(resolution.autGroup));
	194	[ 168, 168 ]
	195	gap> PL:=PartitionsIntoBlockDesigns(rec(
	196	> blockDesign:=DL[2],
	197	> v:=15,blockSizes:=[3],
	198	> tSubsetStructure:=rec(t:=1,lambdas:=[1])));
	199	[ ]
	200	gap> PL:=PartitionsIntoBlockDesigns(rec(
	201	> blockDesign:=DL[3],
	202	> v:=15,blockSizes:=[3],
	203	> tSubsetStructure:=rec(t:=1,lambdas:=[1])));
	204	[ ]
	205	</pre>
	206	<p>
	207	<p>
	208	<h2><a name="SECT002">9.2 Computing resolutions</a></h2>
	209	<p><p>
	210	<a name = "SSEC002.1"></a>
	211	<li><code>MakeResolutionsComponent( </code><var>D</var><code> )</code>
	212	<li><code>MakeResolutionsComponent( </code><var>D</var><code>, </code><var>isolevel</var><code> )</code>
	213	<p>
	214	This function computes resolutions of the block design <var>D</var>, and stores
	215	the result in <code></code><var>D</var><code>.resolutions</code>. If <code></code><var>D</var><code>.resolutions</code> already exists
	216	then it is ignored and overwritten. This function returns no value.
	217	<p>
	218	A <strong>resolution</strong> of a block design <i>D</i> is a partition of the blocks into
	219	subsets, each of which forms a partition of the point set. We say that
	220	two resolutions <i>R</i> and <i>S</i> of <i>D</i> are <strong>isomorphic</strong> if there is an element
	221	<i>g</i> in the automorphism group of <i>D</i>, such that the <i>g</i>-image of <i>R</i>
	222	is <i>S</i>. (Isomorphism defines an equivalence relation on the set of
	223	resolutions of <i>D</i>.)
	224	<p>
	225	The parameter <var>isolevel</var> (default 2) determines how many resolutions are
	226	computed: <var>isolevel</var>=2 means to classify up to isomorphism, <var>isolevel</var>=1
	227	means to determine at least one representative from each isomorphism
	228	class, and <var>isolevel</var>=0 means to determine whether or not <var>D</var> has
	229	a resolution.
	230	<p>
	231	When this function is finished, <code></code><var>D</var><code>.resolutions</code> will have the following
	232	three components:
	233	<p>
	234	<code>list</code>: a list of distinct partitions into block designs forming resolutions
	235	of <var>D</var>;
	236	<p>
	237	<code>pairwiseNonisomorphic</code>: <code>true</code>, <code>false</code> or <code>"unknown"</code>, depending on the
	238	resolutions in <code>list</code> and what is known. If <var>isolevel</var>=0 or <var>isolevel</var>=2
	239	then this component will be <code>true</code>;
	240	<p>
	241	<code>allClassesRepresented</code>: <code>true</code>, <code>false</code> or <code>"unknown"</code>, depending on the
	242	resolutions in <code>list</code> and what is known. If <var>isolevel</var>=1 or <var>isolevel</var>=2
	243	then this component will be <code>true</code>.
	244	<p>
	245	Note that <code></code><var>D</var><code>.resolutions</code> may be changed to contain more information
	246	as a side-effect of other functions in the DESIGN package.
	247	<p>
	248	<pre>
	249	gap> L:=BlockDesigns(rec(v:=9,blockSizes:=[3],
	250	> tSubsetStructure:=rec(t:=2,lambdas:=[1])));;
	251	gap> D:=L[1];;
	252	gap> MakeResolutionsComponent(D);
	253	gap> D;
	254	rec( isBlockDesign := true, v := 9,
	255	blocks := [ [ 1, 2, 3 ], [ 1, 4, 5 ], [ 1, 6, 7 ], [ 1, 8, 9 ],
	256	[ 2, 4, 6 ], [ 2, 5, 8 ], [ 2, 7, 9 ], [ 3, 4, 9 ], [ 3, 5, 7 ],
	257	[ 3, 6, 8 ], [ 4, 7, 8 ], [ 5, 6, 9 ] ],
	258	tSubsetStructure := rec( t := 2, lambdas := [ 1 ] ), isBinary := true,
	259	isSimple := true, blockSizes := [ 3 ], blockNumbers := [ 12 ], r := 4,
	260	autGroup := Group([ (1,2)(5,6)(7,8), (1,3,2)(4,8,7)(5,6,9), (1,2)(4,7)(5,9),
	261	(1,2)(4,9)(5,7)(6,8), (1,4,8,6,9,2)(3,5,7) ]),
	262	resolutions := rec( list := [ rec( partition :=
	263	[ rec( isBlockDesign := true, v := 9,
	264	blocks := [ [ 1, 2, 3 ], [ 4, 7, 8 ], [ 5, 6, 9 ] ] ),
	265	rec( isBlockDesign := true, v := 9,
	266	blocks := [ [ 1, 4, 5 ], [ 2, 7, 9 ], [ 3, 6, 8 ] ] ),
	267	rec( isBlockDesign := true, v := 9,
	268	blocks := [ [ 1, 6, 7 ], [ 2, 5, 8 ], [ 3, 4, 9 ] ] ),
	269	rec( isBlockDesign := true, v := 9,
	270	blocks := [ [ 1, 8, 9 ], [ 2, 4, 6 ], [ 3, 5, 7 ] ] ) ],
	271	autGroup := Group(
	272	[ (2,3)(4,5)(6,7)(8,9), (1,3,2)(4,8,7)(5,6,9),
	273	(1,8,9)(2,4,6)(3,7,5), (1,2)(5,6)(7,8), (1,2)(4,7)(5,9),
	274	(1,2,9,6,8,4)(3,7,5) ]) ) ], pairwiseNonisomorphic := true,
	275	allClassesRepresented := true ) )
	276	</pre>
	277	<p>
	278	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP008.htm">Previous</a>] [<a href ="CHAP010.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
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	0	<html><head><title>[design] 10 XML I/O of block designs</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP009.htm">Previous</a>] [<a href = "theindex.htm">Index</a>]
	3	<h1>10 XML I/O of block designs</h1><p>
	4	<P>
	5	<H3>Sections</H3>
	6	<oL>
	7	<li> <A HREF="CHAP010.htm#SECT001">Writing lists of block designs and their properties in XML-format</a>
	8	<li> <A HREF="CHAP010.htm#SECT002">Reading lists of block designs in XML-format</a>
	9	</ol><p>
	10	<p>
	11	This chapter describes functions to write and read lists of binary block
	12	designs in the <a href="http://designtheory.org">http://designtheory.org</a> external representation
	13	XML-format (see <a href="biblio.htm#Extrep"><cite>Extrep</cite></a>).
	14	<p>
	15	<p>
	16	<h2><a name="SECT001">10.1 Writing lists of block designs and their properties in XML-format</a></h2>
	17	<p><p>
	18	<a name = "SSEC001.1"></a>
	19	<li><code>BlockDesignsToXMLFile( </code><var>filename</var><code>, </code><var>designs</var><code> )</code>
	20	<li><code>BlockDesignsToXMLFile( </code><var>filename</var><code>, </code><var>designs</var><code>, </code><var>include</var><code> )</code>
	21	<li><code>BlockDesignsToXMLFile( </code><var>filename</var><code>, </code><var>designs</var><code>, </code><var>include</var><code>, </code><var>list_id</var><code> )</code>
	22	<p>
	23	This function writes a list of (assumed distinct) binary block designs
	24	(given in DESIGN package format) to a file in external representation
	25	XML-format (version 2.0).
	26	<p>
	27	The parameter <var>filename</var> is a string giving the name of the file, and
	28	<var>designs</var> is a record whose component <code>list</code> contains the list of block
	29	designs (<var>designs</var> can also be a list, in which case it is replaced by
	30	<code>rec(list:=</code><var>designs</var><code>)</code>).
	31	<p>
	32	The record <var>designs</var> should have the following components:
	33	<p>
	34	<code>list</code>: the list of distinct binary block designs in DESIGN package
	35	format;
	36	<p>
	37	<code>pairwiseNonisomorphic</code> (optional): should be <code>true</code> or <code>false</code> or the
	38	string <code>"unknown"</code>, specifying the pairwise-nonisomorphism status of the
	39	designs in <code></code><var>designs</var><code>.list</code>;
	40	<p>
	41	<code>infoXML</code> (optional): should contain a string in XML format for the
	42	<code></code><var>info</var><code></code> element of the <code></code><var>list_of_designs</var><code></code> which is written.
	43	<p>
	44	The combinatorial and group-theoretical properties output for each
	45	design depend on <var>include</var> (default: empty list), which should
	46	be a list containing zero or more of the strings <code>"indicators"</code>,
	47	<code>"resolvable"</code>, <code>"combinatorial_properties"</code>, <code>"automorphism_group"</code>, and
	48	<code>"resolutions"</code>. A shorthand for the list containing all these strings
	49	is <code>"all"</code>. The strings <code>"indicators"</code>, <code>"combinatorial_properties"</code>,
	50	<code>"automorphism_group"</code>, and <code>"resolutions"</code> are used to specify that
	51	those subtrees of the external representation of each design are to
	52	be expanded and written out. In the case of <code>"resolutions"</code> being in
	53	<var>include</var>, <strong>all</strong> resolutions up to isomorphism will be determined and
	54	written out. The string <code>"resolvable"</code> is used to specify that the
	55	<code>resolvable</code> indicator must be set (usually this is not forced), if
	56	the <code>indicators</code> subtree is written out, and also that if a design is
	57	resolvable but <code>"resolutions"</code> is not in <var>include</var>, then one and only
	58	one resolution should be written out in the <code>resolutions</code> subtree.
	59	<p>
	60	If <var>list_id</var> is given then the id's of the output designs will be
	61	<code></code><var>list_id</var><code>-0</code>, <code></code><var>list_id</var><code>-1</code>, <code></code><var>list_id</var><code>-2</code>, ...
	62	<p>
	63	<pre>
	64	gap> D:=[ BlockDesign(3, [[1,2],[1,3]]),
	65	> BlockDesign(3, [[1,2],[1,2],[2,3]]) ];;
	66	gap> designs:=rec(list:=D, pairwiseNonisomorphic:=true);;
	67	gap> BlockDesignsToXMLFile("example.xml",designs,[],"example");
	68	</pre>
	69	<p>
	70	<p>
	71	<h2><a name="SECT002">10.2 Reading lists of block designs in XML-format</a></h2>
	72	<p><p>
	73	<a name = "SSEC002.1"></a>
	74	<li><code>BlockDesignsFromXMLFile( </code><var>filename</var><code> )</code>
	75	<p>
	76	This function reads a file with name <var>filename</var>, containing a list of
	77	distinct binary block designs in external representation XML-format,
	78	and returns a record <var>designs</var> in DESIGN package format containing
	79	the essential information in this file.
	80	<p>
	81	The record <var>designs</var> contains the following components:
	82	<p>
	83	<code>list</code>: a list of block designs in DESIGN package format of
	84	the list of block designs in the file (certain elements such as
	85	<code></code><var>statistical_properties</var><code></code> are stored verbatim as strings; certain other
	86	elements are not stored since it is usually easier and more reliable to
	87	recompute them -- this can be done when the block designs are written
	88	out in XML format);
	89	<p>
	90	<code>pairwiseNonisomorphic</code> is set according to the attribute
	91	<code>pairwise_nonisomorphic</code> of the XML element <code></code><var>list_of_designs</var><code></code>.
	92	The component <code>pairwiseNonisomorphic</code> is <code>false</code> if this attribute
	93	is <code>false</code>, <code>true</code> if this attribute is <code>true</code>, and <code>"unknown"</code> otherwise;
	94	<p>
	95	<code>infoXML</code> is bound iff the <code></code><var>info</var><code></code> element occurs as a child of the
	96	XML <code></code><var>list_of_designs</var><code></code> element, and if bound, contains this <code></code><var>info</var><code></code>
	97	element in a string.
	98	<p>
	99	<pre>
	100	gap> BlockDesignsFromXMLFile("example.xml");
	101	rec(
	102	infoXML := "<info>\n<software>\n[ DESIGN-1.7, GRAPE-4.8.2, GAPDoc-1.6.2, GAP\
	103	-4.10.1 ]\n</software>\n</info>",
	104	list :=
	105	[
	106	rec( blocks := [ [ 1, 2 ], [ 1, 3 ] ], id := "example-0",
	107	isBinary := true, isBlockDesign := true, v := 3 ),
	108	rec( blocks := [ [ 1, 2 ], [ 1, 2 ], [ 2, 3 ] ], id := "example-1",
	109	isBinary := true, isBlockDesign := true, v := 3 ) ],
	110	pairwiseNonisomorphic := true )
	111	</pre>
	112	<p>
	113	[<a href = "chapters.htm">Up</a>] [<a href ="CHAP009.htm">Previous</a>] [<a href = "theindex.htm">Index</a>]
	114	<P>
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	0	<html><head><title>design : a GAP 4 package - References</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - References</h1><dl>
	3
	4	<dt><a name="BaCh"><b>[BaCh]</b></a><dd>
	5	R. A. Bailey and P. E. Chigbu.
	6	<br> Enumeration of semi-latin squares.
	7	<br> <em>Discrete Math.</em>, 167-168:73--84, 1997.
	8
	9	<dt><a name="BaCa"><b>[BaCa]</b></a><dd>
	10	R. A. Bailey and P. J. Cameron.
	11	<br> Combinatorics of optimal designs.
	12	<br> In S. Huczynska, J. D. Mitchell, and C. M. Roney-Dougal, editors,
	13	<em>Surveys in Combinatorics 2009</em>, volume 365 of <em>London Math. Soc.
	14	Lecture Notes</em>, pages 19--73. Cambridge University Press, 2009.
	15
	16	<dt><a name="Dotw"><b>[Dotw]</b></a><dd>
	17	R. A. Bailey, P. J. Cameron, P. Dobcsányi, J. P. Morgan, and L. H. Soicher.
	18	<br> Designs on the web.
	19	<br> <em>Discrete Math.</em>, 306:3014--3027, 2006.
	20	<br> <a href="http://dx.doi.org/10.1016/j.disc.2004.10.027">http://dx.doi.org/10.1016/j.disc.2004.10.027</a>.
	21
	22	<dt><a name="BaRo"><b>[BaRo]</b></a><dd>
	23	R. A. Bailey and G. Royle.
	24	<br> Optimal semi-latin squares with side six and block size two.
	25	<br> <em>Proc. Roy. Soc. London, Ser. A</em>, 453:1903--1914, 1997.
	26
	27	<dt><a name="Extrep"><b>[Extrep]</b></a><dd>
	28	P. J. Cameron, P. Dobcsányi, J. P. Morgan, and L. H. Soicher.
	29	<br> <em>The external representation of block designs, Version 2.0</em>,
	30	2004.
	31	<br> <a href="http://designtheory.org/library/extrep/">http://designtheory.org/library/extrep/</a>.
	32
	33	<dt><a name="CaSo"><b>[CaSo]</b></a><dd>
	34	P. J. Cameron and L. H. Soicher.
	35	<br> Block intersection polynomials.
	36	<br> <em>Bull. London Math. Soc.</em>, 39:559--564, 2007.
	37	<br> <a href="http://dx.doi.org/10.1112/blms/bdm034">http://dx.doi.org/10.1112/blms/bdm034</a>.
	38
	39	<dt><a name="JK07"><b>[JK07]</b></a><dd>
	40	Tommi Juntilla and Petteri Kaski.
	41	<br> Engineering an efficient canonical labeling tool for large and sparse
	42	graphs.
	43	<br> In David Applegate et al., editor, <em>Proceedings of the Ninth
	44	Workshop on Algorithm Engineering and Experiments and the Fourth Workshop on
	45	Analytic Algorithmics and Combinatorics</em>, pages 135--149. SIAM, 2007.
	46	<br> bliss homepage: <a href="http://www.tcs.hut.fi/Software/bliss/">http://www.tcs.hut.fi/Software/bliss/</a>.
	47
	48	<dt><a name="Nau90"><b>[Nau90]</b></a><dd>
	49	Brendan D. McKay.
	50	<br> <em>nauty user's guide (version 1.5), Technical report
	51	TR-CS-90-02</em>.
	52	<br> Australian National University, Computer Science Department, 1990.
	53	<br> nauty homepage: <a href="http://cs.anu.edu.au/people/bdm/nauty/">http://cs.anu.edu.au/people/bdm/nauty/</a>.
	54
	55	<dt><a name="MP14"><b>[MP14]</b></a><dd>
	56	Brendan D. McKay and Adolfo Piperno.
	57	<br> Practical graph isomorphism, ii.
	58	<br> <em>J. Symbolic Comput.</em>, 60:94--112, 2014.
	59
	60	<dt><a name="McSo"><b>[McSo]</b></a><dd>
	61	J. P. McSorley and L. H. Soicher.
	62	<br> Constructing <i>t</i>-designs from <i>t</i>-wise balanced designs.
	63	<br> <em>European J. Combinatorics</em>, 28:567--571, 2007.
	64	<br> <a href="http://dx.doi.org/10.1016/j.ejc.2005.02.003">http://dx.doi.org/10.1016/j.ejc.2005.02.003</a>.
	65
	66	<dt><a name="Soi1"><b>[Soi1]</b></a><dd>
	67	L. H. Soicher.
	68	<br> More on block intersection polynomials and new applications to graphs
	69	and block designs.
	70	<br> <em>J. Comb. Theory, Ser. A</em>, 117:799--809, 2010.
	71	<br> <a href="http://dx.doi.org/10.1016/j.jcta.2010.03.005">http://dx.doi.org/10.1016/j.jcta.2010.03.005</a>.
	72
	73	<dt><a name="Grape"><b>[Grape]</b></a><dd>
	74	L. H. Soicher.
	75	<br> <em>The GRAPE package for GAP, Version 4.8.2</em>, 2019.
	76	<br> <a href="https://gap-packages.github.io/grape">https://gap-packages.github.io/grape</a>.
	77
	78	</dl><p>
	79	[<a href="chapters.htm">Up</a>]<p>
	80	<P>
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	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Chapters</h1>
	3	<ul>
	4	<li><a href="theindex.htm">Index</a>
	5	</ul>
	6	<ol>
	7	<li><a href="CHAP001.htm">Design</a>
	8	<li><a href="CHAP002.htm">Information from block design parameters</a>
	9	<li><a href="CHAP003.htm">Constructing block designs</a>
	10	<li><a href="CHAP004.htm">Determining basic properties of block designs</a>
	11	<li><a href="CHAP005.htm">Matrices and efficiency measures for block designs</a>
	12	<li><a href="CHAP006.htm">Automorphism groups and isomorphism testing for block designs</a>
	13	<li><a href="CHAP007.htm">Classifying block designs</a>
	14	<li><a href="CHAP008.htm">Classifying semi-Latin squares</a>
	15	<li><a href="CHAP009.htm">Partitioning block designs</a>
	16	<li><a href="CHAP010.htm">XML I/O of block designs</a>
	17	</ol>
	18	<ul>
	19	<li><a href="biblio.htm">References</a>
	20	<li><a href="theindex.htm">Index</a>
	21	</ul><p>
	22	<P>
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	0	<html><head><title>design : a GAP 4 package - Index A</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index A</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>AddedBlocksBlockDesign <a href="CHAP003.htm#SSEC001.10">3.1.10</a>
	22	<dt>AddedPointBlockDesign <a href="CHAP003.htm#SSEC001.9">3.1.9</a>
	23	<dt>AffineResolvableMu <a href="CHAP004.htm#SSEC001.15">4.1.15</a>
	24	<dt>AGPointFlatBlockDesign <a href="CHAP003.htm#SSEC001.2">3.1.2</a>
	25	<dt>AllTDesignLambdas <a href="CHAP004.htm#SSEC001.14">4.1.14</a>
	26	<dt>AutGroupBlockDesign <a href="CHAP006.htm#SSEC001.1">6.1.1</a>
	27	<dt>Automorphism groups and isomorphism testing for block designs <a href="CHAP006.htm">6.0</a>
	28	</dl><p>
	29	[<a href="chapters.htm">Up</a>]<p>
	30	<P>
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	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index B</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>binary block design <a href="CHAP001.htm#I1">1.3</a>
	22	<dt>block design <a href="CHAP001.htm#I0">1.3</a>
	23	<dt>Block intersection polynomials <a href="CHAP002.htm#SECT002">2.2</a>
	24	<dt>BlockDesign <a href="CHAP003.htm#SSEC001.1">3.1.1</a>
	25	<dt>BlockDesignBlocks <a href="CHAP004.htm#SSEC001.7">4.1.7</a>
	26	<dt>BlockDesignEfficiency <a href="CHAP005.htm#SSEC002.1">5.2.1</a>
	27	<dt>BlockDesignIsomorphismClassRepresentatives <a href="CHAP006.htm#SSEC002.2">6.2.2</a>
	28	<dt>BlockDesignPoints <a href="CHAP004.htm#SSEC001.5">4.1.5</a>
	29	<dt>BlockDesigns <a href="CHAP007.htm#SSEC001.1">7.1.1</a>
	30	<dt>BlockDesignsFromXMLFile <a href="CHAP010.htm#SSEC002.1">10.2.1</a>
	31	<dt>BlockDesignsToXMLFile <a href="CHAP010.htm#SSEC001.1">10.1.1</a>
	32	<dt>BlockIntersectionPolynomial <a href="CHAP002.htm#SSEC002.1">2.2.1</a>
	33	<dt>BlockIntersectionPolynomialCheck <a href="CHAP002.htm#SSEC002.2">2.2.2</a>
	34	<dt>BlockNumbers <a href="CHAP004.htm#SSEC001.10">4.1.10</a>
	35	<dt>BlockSizes <a href="CHAP004.htm#SSEC001.9">4.1.9</a>
	36	</dl><p>
	37	[<a href="chapters.htm">Up</a>]<p>
	38	<P>
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	0	<html><head><title>design : a GAP 4 package - Index C</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index C</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>Classifying block designs <a href="CHAP007.htm">7.0</a>
	22	<dt>Classifying semi-Latin squares <a href="CHAP008.htm">8.0</a>
	23	<dt>ComplementBlocksBlockDesign <a href="CHAP003.htm#SSEC001.6">3.1.6</a>
	24	<dt>Computing an interval for a certain real zero of a rational polynomial <a href="CHAP005.htm#SECT003">5.3</a>
	25	<dt>Computing automorphism groups <a href="CHAP006.htm#SECT001">6.1</a>
	26	<dt>Computing resolutions <a href="CHAP009.htm#SECT002">9.2</a>
	27	<dt>ConcurrenceMatrix <a href="CHAP005.htm#SSEC001.2">5.1.2</a>
	28	<dt>Constructing block designs <a href="CHAP003.htm">3.0</a>
	29	</dl><p>
	30	[<a href="chapters.htm">Up</a>]<p>
	31	<P>
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	0	<html><head><title>design : a GAP 4 package - Index D</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index D</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>DeletedBlocksBlockDesign <a href="CHAP003.htm#SSEC001.8">3.1.8</a>
	22	<dt>DeletedPointsBlockDesign <a href="CHAP003.htm#SSEC001.7">3.1.7</a>
	23	<dt>derived design <a href="CHAP003.htm#I0">3.1</a>
	24	<dt>DerivedBlockDesign <a href="CHAP003.htm#SSEC001.11">3.1.11</a>
	25	<dt>Design <a href="CHAP001.htm">1.0</a>
	26	<dt>DESIGN_IntervalForLeastRealZero <a href="CHAP005.htm#SSEC003.1">5.3.1</a>
	27	<dt>Determining basic properties of block designs <a href="CHAP004.htm">4.0</a>
	28	<dt>DualBlockDesign <a href="CHAP003.htm#SSEC001.5">3.1.5</a>
	29	</dl><p>
	30	[<a href="chapters.htm">Up</a>]<p>
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	0	<html><head><title>design : a GAP 4 package - Index E</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index E</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>Example of the use of DESIGN <a href="CHAP001.htm#SECT004">1.4</a>
	22	</dl><p>
	23	[<a href="chapters.htm">Up</a>]<p>
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	0	<html><head><title>design : a GAP 4 package - Index F</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index F</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>Functions to construct block designs <a href="CHAP003.htm#SECT001">3.1</a>
	22	</dl><p>
	23	[<a href="chapters.htm">Up</a>]<p>
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	0	<html><head><title>design : a GAP 4 package - Index I</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index I</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>Information from $t$-design parameters <a href="CHAP002.htm#SECT001">2.1</a>
	22	<dt>Information from block design parameters <a href="CHAP002.htm">2.0</a>
	23	<dt>InformationMatrix <a href="CHAP005.htm#SSEC001.3">5.1.3</a>
	24	<dt>Installing the DESIGN Package <a href="CHAP001.htm#SECT001">1.1</a>
	25	<dt>IsBinaryBlockDesign <a href="CHAP004.htm#SSEC001.2">4.1.2</a>
	26	<dt>IsBlockDesign <a href="CHAP004.htm#SSEC001.1">4.1.1</a>
	27	<dt>IsConnectedBlockDesign <a href="CHAP004.htm#SSEC001.4">4.1.4</a>
	28	<dt>IsIsomorphicBlockDesign <a href="CHAP006.htm#SSEC002.1">6.2.1</a>
	29	<dt>IsSimpleBlockDesign <a href="CHAP004.htm#SSEC001.3">4.1.3</a>
	30	</dl><p>
	31	[<a href="chapters.htm">Up</a>]<p>
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	0	<html><head><title>design : a GAP 4 package - Index L</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index L</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>Loading DESIGN <a href="CHAP001.htm#SECT002">1.2</a>
	22	</dl><p>
	23	[<a href="chapters.htm">Up</a>]<p>
	24	<P>
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	0	<html><head><title>design : a GAP 4 package - Index M</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index M</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>MakeResolutionsComponent <a href="CHAP009.htm#SSEC002.1">9.2.1</a>
	22	<dt>Matrices and efficiency measures for block designs <a href="CHAP005.htm">5.0</a>
	23	<dt>Matrices associated with a block design <a href="CHAP005.htm#SECT001">5.1</a>
	24	</dl><p>
	25	[<a href="chapters.htm">Up</a>]<p>
	26	<P>
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	0	<html><head><title>design : a GAP 4 package - Index N</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index N</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>NrBlockDesignBlocks <a href="CHAP004.htm#SSEC001.8">4.1.8</a>
	22	<dt>NrBlockDesignPoints <a href="CHAP004.htm#SSEC001.6">4.1.6</a>
	23	</dl><p>
	24	[<a href="chapters.htm">Up</a>]<p>
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	0	<html><head><title>design : a GAP 4 package - Index P</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index P</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>PairwiseBalancedLambda <a href="CHAP004.htm#SSEC001.12">4.1.12</a>
	22	<dt>Partitioning a block design into block designs <a href="CHAP009.htm#SECT001">9.1</a>
	23	<dt>Partitioning block designs <a href="CHAP009.htm">9.0</a>
	24	<dt>PartitionsIntoBlockDesigns <a href="CHAP009.htm#SSEC001.1">9.1.1</a>
	25	<dt>PGPointFlatBlockDesign <a href="CHAP003.htm#SSEC001.3">3.1.3</a>
	26	<dt>PointBlockIncidenceMatrix <a href="CHAP005.htm#SSEC001.1">5.1.1</a>
	27	</dl><p>
	28	[<a href="chapters.htm">Up</a>]<p>
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	0	<html><head><title>design : a GAP 4 package - Index R</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index R</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>Reading lists of block designs in XML-format <a href="CHAP010.htm#SECT002">10.2</a>
	22	<dt>ReplicationNumber <a href="CHAP004.htm#SSEC001.11">4.1.11</a>
	23	<dt>residual design <a href="CHAP003.htm#I1">3.1</a>
	24	<dt>ResidualBlockDesign <a href="CHAP003.htm#SSEC001.12">3.1.12</a>
	25	<dt>ResolvableTDesignBlockMultiplicityBound <a href="CHAP002.htm#SSEC001.6">2.1.6</a>
	26	</dl><p>
	27	[<a href="chapters.htm">Up</a>]<p>
	28	<P>
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	0	<html><head><title>design : a GAP 4 package - Index S</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index S</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>semi-Latin square <a href="CHAP008.htm#I0">8.1</a>
	22	<dt>Semi-Latin squares and SOMAs <a href="CHAP008.htm#SECT001">8.1</a>
	23	<dt>SemiLatinSquareDuals <a href="CHAP008.htm#SSEC002.1">8.2.1</a>
	24	<dt>SOMA <a href="CHAP008.htm#I1">8.1</a>
	25	<dt>SteinerSystemIntersectionTriangle <a href="CHAP002.htm#SSEC001.4">2.1.4</a>
	26	</dl><p>
	27	[<a href="chapters.htm">Up</a>]<p>
	28	<P>
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	0	<html><head><title>design : a GAP 4 package - Index T</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index T</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>t-design <a href="CHAP002.htm#I0">2.1</a>
	22	<dt>TDesignBlockMultiplicityBound <a href="CHAP002.htm#SSEC001.5">2.1.5</a>
	23	<dt>TDesignFromTBD <a href="CHAP003.htm#SSEC001.13">3.1.13</a>
	24	<dt>TDesignIntersectionTriangle <a href="CHAP002.htm#SSEC001.3">2.1.3</a>
	25	<dt>TDesignLambdaMin <a href="CHAP002.htm#SSEC001.2">2.1.2</a>
	26	<dt>TDesignLambdas <a href="CHAP002.htm#SSEC001.1">2.1.1</a>
	27	<dt>Testing isomorphism <a href="CHAP006.htm#SECT002">6.2</a>
	28	<dt>The function BlockDesignEfficiency <a href="CHAP005.htm#SECT002">5.2</a>
	29	<dt>The function BlockDesigns <a href="CHAP007.htm#SECT001">7.1</a>
	30	<dt>The function SemiLatinSquareDuals <a href="CHAP008.htm#SECT002">8.2</a>
	31	<dt>The functions for basic properties <a href="CHAP004.htm#SECT001">4.1</a>
	32	<dt>The structure of a block design in DESIGN <a href="CHAP001.htm#SECT003">1.3</a>
	33	<dt>TSubsetLambdasVector <a href="CHAP004.htm#SSEC001.13">4.1.13</a>
	34	</dl><p>
	35	[<a href="chapters.htm">Up</a>]<p>
	36	<P>
	37	<address>design manual<br>March 2019
	38	</address></body></html>⏎

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htm/indxW.htm less more

	0	<html><head><title>design : a GAP 4 package - Index W</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index W</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>WittDesign <a href="CHAP003.htm#SSEC001.4">3.1.4</a>
	22	<dt>Writing lists of block designs and their properties in XML-format <a href="CHAP010.htm#SECT001">10.1</a>
	23	</dl><p>
	24	[<a href="chapters.htm">Up</a>]<p>
	25	<P>
	26	<address>design manual<br>March 2019
	27	</address></body></html>⏎

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htm/indxX.htm less more

	0	<html><head><title>design : a GAP 4 package - Index X</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index X</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	<dt>XML I/O of block designs <a href="CHAP010.htm">10.0</a>
	22	</dl><p>
	23	[<a href="chapters.htm">Up</a>]<p>
	24	<P>
	25	<address>design manual<br>March 2019
	26	</address></body></html>⏎

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htm/theindex.htm less more

	0	<html><head><title>design : a GAP 4 package - Index _</title></head>
	1	<body text="#000000" bgcolor="#ffffff">
	2	<h1><font face="Gill Sans,Helvetica,Arial">design</font> : a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package - Index _</h1>
	3	<p>
	4	<a href="theindex.htm">_</A>
	5	<a href="indxA.htm">A</A>
	6	<a href="indxB.htm">B</A>
	7	<a href="indxC.htm">C</A>
	8	<a href="indxD.htm">D</A>
	9	<a href="indxE.htm">E</A>
	10	<a href="indxF.htm">F</A>
	11	<a href="indxI.htm">I</A>
	12	<a href="indxL.htm">L</A>
	13	<a href="indxM.htm">M</A>
	14	<a href="indxN.htm">N</A>
	15	<a href="indxP.htm">P</A>
	16	<a href="indxR.htm">R</A>
	17	<a href="indxS.htm">S</A>
	18	<a href="indxT.htm">T</A>
	19	<a href="indxW.htm">W</A>
	20	<a href="indxX.htm">X</A>
	21	</dl><p>
	22	[<a href="chapters.htm">Up</a>]<p>
	23	<P>
	24	<address>design manual<br>March 2019
	25	</address></body></html>⏎

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lib/blockdesign.g less more

0	0	############################################################################
1	1	##
2		## blockdesign.g Design 1.6 Package Leonard Soicher
	2	## blockdesign.g Design 1.7 Package Leonard Soicher
3	3	##
4	4	##
5	5	# Functions to study, construct and classify (sub)block designs

7	7	# block designs with given properties.
8	8	# At present there is limited support for non-binary block designs.
9	9	#
10		# Copyright (C) 2003-2011 Leonard H. Soicher
	10	# Copyright (C) 2003-2019 Leonard H. Soicher
11	11	#
12	12	# This program is free software; you can redistribute it and/or modify
13	13	# it under the terms of the GNU General Public License as published by

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lib/blockdesign_io.g less more

0	0	#############################################################################
1	1	##
2		## blockdesign_io.g Design 1.6 Package Leonard Soicher
	2	## blockdesign_io.g Design 1.7 Package Leonard Soicher
3	3	##
4	4	##
5	5	# The "GAP Expander/Writer" to expand the information about

10	10	# DTRS external representation XML-format, and to convert these
11	11	# block designs into GAP-format.
12	12	#
13		# Copyright (C) 2003-2011 Leonard H. Soicher
	13	# Copyright (C) 2003-2019 Leonard H. Soicher
14	14	#
15	15	# This program is free software; you can redistribute it and/or modify
16	16	# it under the terms of the GNU General Public License as published by

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tst/testall.g less more

	0	LoadPackage( "design" );
	1
	2	TestDirectory(DirectoriesPackageLibrary( "design", "tst" ),
	3	rec(exitGAP := true,
	4	testOptions := rec(compareFunction := "uptowhitespace") ) );
	5
	6	FORCE_QUIT_GAP(1); # if we ever get here, there was an error

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tst/testall.tst less more

	0	#############################################################################
	1	##
	2	## testall.tst DESIGN package Leonard Soicher
	3	##
	4	## To create a test file, place GAP prompts, input and output exactly as
	5	## they must appear in the GAP session. Do not remove lines containing
	6	## START_TEST and STOP_TEST statements.
	7	##
	8	## The first line starts the test. START_TEST reinitializes the caches and
	9	## the global random number generator, in order to be independent of the
	10	## reading order of several test files. Furthermore, the assertion level
	11	## is set to 2 by START_TEST and set back to the previous value in the
	12	## subsequent STOP_TEST call.
	13	##
	14	## The argument of STOP_TEST may be an arbitrary identifier string.
	15	##
	16	gap> START_TEST("DESIGN package: testall.tst");
	17
	18	# Note that you may use comments in the test file
	19	# and also separate parts of the test by empty lines
	20
	21	# First load the package without banner (the banner must be suppressed to
	22	# avoid reporting discrepancies in the case when the package is already
	23	# loaded)
	24	gap> LoadPackage("design",false);
	25	true
	26	gap> H:=CyclicGroup(IsPermGroup,20);;
	27	gap> D:=BlockDesigns(rec(v:=21,blockSizes:=[4,5],
	28	> tSubsetStructure:=rec(t:=2,lambdas:=[1]),
	29	> requiredAutSubgroup:=H ));;
	30	gap> Length(D);
	31	1
	32	gap> D:=D[1];;
	33	gap> BlockSizes(D);
	34	[ 4, 5 ]
	35	gap> BlockNumbers(D);
	36	[ 20, 9 ]
	37	gap> Size(AutGroupBlockDesign(D));
	38	80
	39	gap> Dstar:=TDesignFromTBD(D,2,4);;
	40	gap> AllTDesignLambdas(Dstar);
	41	[ 105, 20, 3 ]
	42	gap> IsSimpleBlockDesign(Dstar);
	43	false
	44	gap> Size(AutGroupBlockDesign(Dstar));
	45	80
	46	gap> near_resolutions:=PartitionsIntoBlockDesigns(rec(
	47	> blockDesign:=Dstar,
	48	> v:=21,blockSizes:=[4],
	49	> tSubsetStructure:=rec(t:=0,lambdas:=[5]),
	50	> blockIntersectionNumbers:=[[ [0] ]],
	51	> requiredAutSubgroup:=SylowSubgroup(H,5) ));;
	52	gap> Length(near_resolutions);
	53	2
	54	gap> Collected(List(near_resolutions,x->Size(x.autGroup)));
	55	[ [ 5, 1 ], [ 20, 1 ] ]
	56	gap> TDesignBlockMultiplicityBound(2,21,4,3);
	57	3
	58	gap> Collected(List(Collected(BlockDesignBlocks(Dstar)),x->x[2]));
	59	[ [ 1, 45 ], [ 3, 20 ] ]
	60	gap> TDesignLambdas(5,24,8,1);
	61	[ 759, 253, 77, 21, 5, 1 ]
	62	gap> TDesignLambdaMin(5,24,8);
	63	1
	64	gap> TDesignLambdaMin(2,12,4);
	65	3
	66	gap> TDesignLambdas(2,12,4,3);
	67	[ 33, 11, 3 ]
	68	gap> TDesignIntersectionTriangle(2,12,4,3);
	69	[ [ 33, 22, 14 ], [ 11, 8 ], [ 3 ] ]
	70	gap> TDesignLambdas(2,12,4,2);
	71	fail
	72	gap> TDesignIntersectionTriangle(2,12,4,2);
	73	fail
	74	gap> SteinerSystemIntersectionTriangle(5,24,8);
	75	[ [ 759, 506, 330, 210, 130, 78, 46, 30, 30 ],
	76	[ 253, 176, 120, 80, 52, 32, 16, 0 ], [ 77, 56, 40, 28, 20, 16, 16 ],
	77	[ 21, 16, 12, 8, 4, 0 ], [ 5, 4, 4, 4, 4 ], [ 1, 0, 0, 0 ], [ 1, 0, 0 ],
	78	[ 1, 0 ], [ 1 ] ]
	79	gap> TDesignIntersectionTriangle(5,24,8,1);
	80	[ [ 759, 506, 330, 210, 130, 78 ], [ 253, 176, 120, 80, 52 ],
	81	[ 77, 56, 40, 28 ], [ 21, 16, 12 ], [ 5, 4 ], [ 1 ] ]
	82	gap> TDesignBlockMultiplicityBound(5,16,7,5);
	83	2
	84	gap> TDesignBlockMultiplicityBound(2,36,6,1);
	85	0
	86	gap> TDesignBlockMultiplicityBound(2,36,6,2);
	87	2
	88	gap> TDesignBlockMultiplicityBound(2,15,5,2);
	89	0
	90	gap> TDesignBlockMultiplicityBound(2,15,5,4);
	91	2
	92	gap> TDesignBlockMultiplicityBound(2,11,4,6);
	93	3
	94	gap> ResolvableTDesignBlockMultiplicityBound(5,12,6,1);
	95	1
	96	gap> ResolvableTDesignBlockMultiplicityBound(2,21,7,3);
	97	0
	98	gap> TDesignBlockMultiplicityBound(2,21,7,3);
	99	1
	100	gap> ResolvableTDesignBlockMultiplicityBound(2,12,4,3);
	101	1
	102	gap> TDesignBlockMultiplicityBound(2,12,4,3);
	103	2
	104	gap> x:=Indeterminate(Rationals,1);
	105	x_1
	106	gap> m:=[0,0,0,0,0,0,0,1];;
	107	gap> lambdavec:=TDesignLambdas(6,14,7,4);
	108	[ 1716, 858, 396, 165, 60, 18, 4 ]
	109	gap> B:=BlockIntersectionPolynomial(x,m,lambdavec);
	110	1715x_1^6-10269x_1^5+34685x_1^4-69615x_1^3+84560x_1^2-56196x_1+15120
	111	gap> Value(B,1);
	112	0
	113	gap> m:=[0,0,0,0,0,0,0,1];;
	114	gap> lambdavec:=TDesignLambdas(6,14,7,4);
	115	[ 1716, 858, 396, 165, 60, 18, 4 ]
	116	gap> BlockIntersectionPolynomialCheck(m,lambdavec);
	117	true
	118	gap> m:=[1,0,0,0,0,0,0,1];;
	119	gap> BlockIntersectionPolynomialCheck(m,lambdavec);
	120	false
	121	gap> D:=BlockDesign(7, [[1,2,4]], Group((1,2,3,4,5,6,7)));;
	122	gap> AllTDesignLambdas(D);
	123	[ 7, 3, 1 ]
	124	gap> D:=AGPointFlatBlockDesign(2,4,1);;
	125	gap> AllTDesignLambdas(D);
	126	[ 20, 5, 1 ]
	127	gap> D:=PGPointFlatBlockDesign(3,2,1);;
	128	gap> AllTDesignLambdas(D);
	129	[ 35, 7, 1 ]
	130	gap> W24:=WittDesign(24);;
	131	gap> AllTDesignLambdas(W24);
	132	[ 759, 253, 77, 21, 5, 1 ]
	133	gap> Size(AutomorphismGroup(W24));
	134	244823040
	135	gap> W10:=WittDesign(10);;
	136	gap> AllTDesignLambdas(W10);
	137	[ 30, 12, 4, 1 ]
	138	gap> Size(AutomorphismGroup(W10));
	139	1440
	140	gap> D:=BlockDesign(4,[[1,3],[2,3,4],[3,4]]);;
	141	gap> dualD:=DualBlockDesign(D);;
	142	gap> dualD.blocks;
	143	[ [ 1 ], [ 1, 2, 3 ], [ 2 ], [ 2, 3 ] ]
	144	gap> D:=PGPointFlatBlockDesign(2,2,1);;
	145	gap> AllTDesignLambdas(D);
	146	[ 7, 3, 1 ]
	147	gap> C:=ComplementBlocksBlockDesign(D);;
	148	gap> AllTDesignLambdas(C);
	149	[ 7, 4, 2 ]
	150	gap> D:=BlockDesigns(rec(v:=11,blockSizes:=[5],
	151	> tSubsetStructure:=rec(t:=2,lambdas:=[2])))[1];;
	152	gap> AllTDesignLambdas(D);
	153	[ 11, 5, 2 ]
	154	gap> DP:=DeletedPointsBlockDesign(D,[5,8]);;
	155	gap> PairwiseBalancedLambda(DP);
	156	2
	157	gap> DD:=DerivedBlockDesign(D,6);;
	158	gap> AllTDesignLambdas(DD);
	159	[ 5, 2 ]
	160	gap> DD:=DerivedBlockDesign(D,D.blocks[6]);;
	161	gap> AllTDesignLambdas(DD);
	162	[ 10, 4, 1 ]
	163	gap> RD:=ResidualBlockDesign(D,6);;
	164	gap> AllTDesignLambdas(RD);
	165	[ 6, 3 ]
	166	gap> RD:=ResidualBlockDesign(D,D.blocks[6]);;
	167	gap> AllTDesignLambdas(RD);
	168	[ 10, 5, 2 ]
	169	gap> D:=BlockDesigns(rec(v:=10, blockSizes:=[3,4],
	170	> tSubsetStructure:=rec(t:=2,lambdas:=[1])))[1];;
	171	gap> PairwiseBalancedLambda(D);
	172	1
	173	gap> Dstar:=TDesignFromTBD(D,2,3);;
	174	gap> AllTDesignLambdas(Dstar);
	175	[ 30, 9, 2 ]
	176	gap> IsBlockDesign(5);
	177	false
	178	gap> IsBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) );
	179	true
	180	gap> IsBinaryBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) );
	181	true
	182	gap> IsBinaryBlockDesign( BlockDesign(2,[[1],[1,2],[1,2,2]]) );
	183	false
	184	gap> IsSimpleBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) );
	185	false
	186	gap> IsSimpleBlockDesign( BlockDesign(2,[[1],[1,2],[1,2,2]]) );
	187	true
	188	gap> IsConnectedBlockDesign( BlockDesign(2,[[1],[2]]) );
	189	false
	190	gap> IsConnectedBlockDesign( BlockDesign(2,[[1,2]]) );
	191	true
	192	gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]);;
	193	gap> Length(BlockDesignPoints(D));
	194	3
	195	gap> NrBlockDesignPoints(D);
	196	3
	197	gap> BlockDesignBlocks(D);
	198	[ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ]
	199	gap> NrBlockDesignBlocks(D);
	200	4
	201	gap> BlockSizes( BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]) );
	202	[ 1, 3 ]
	203	gap> D:=BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]);;
	204	gap> BlockSizes(D);
	205	[ 1, 3 ]
	206	gap> BlockNumbers(D);
	207	[ 3, 2 ]
	208	gap> ReplicationNumber(BlockDesign(4,[[1],[1,2],[2,3,3],[4,4]]));
	209	2
	210	gap> ReplicationNumber(BlockDesign(4,[[1],[1,2],[2,3],[4,4]]));
	211	fail
	212	gap> D:=BlockDesigns(rec(v:=10, blockSizes:=[3,4],
	213	> tSubsetStructure:=rec(t:=2,lambdas:=[1])))[1];;
	214	gap> PairwiseBalancedLambda(D);
	215	1
	216	gap> D:=BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]);;
	217	gap> TSubsetLambdasVector(D,0);
	218	[ 5 ]
	219	gap> TSubsetLambdasVector(D,1);
	220	[ 3, 4, 2 ]
	221	gap> TSubsetLambdasVector(D,2);
	222	[ 3, 1, 1 ]
	223	gap> TSubsetLambdasVector(D,3);
	224	[ 1 ]
	225	gap> AllTDesignLambdas(PGPointFlatBlockDesign(3,2,1));
	226	[ 35, 7, 1 ]
	227	gap> P:=PGPointFlatBlockDesign(2,3,1);;
	228	gap> AffineResolvableMu(P);
	229	fail
	230	gap> A:=ResidualBlockDesign(P,P.blocks[1]);;
	231	gap> AffineResolvableMu(A);
	232	1
	233	gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));;
	234	gap> PointBlockIncidenceMatrix(D);
	235	[ [ 1, 1, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 1, 1, 0, 0, 0, 0 ],
	236	[ 1, 0, 0, 0, 0, 1, 1, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 1, 1 ],
	237	[ 0, 1, 0, 1, 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 1, 1, 0 ],
	238	[ 0, 1, 0, 0, 1, 0, 0, 0, 1 ], [ 0, 0, 1, 1, 0, 0, 0, 1, 0 ],
	239	[ 0, 0, 1, 0, 1, 0, 1, 0, 0 ], [ 0, 0, 1, 0, 0, 1, 0, 0, 1 ],
	240	[ 0, 0, 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 0, 1, 1, 0, 1, 0 ] ]
	241	gap> ConcurrenceMatrix(D);
	242	[ [ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0 ],
	243	[ 1, 3, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1 ],
	244	[ 1, 1, 3, 1, 1, 1, 0, 0, 1, 1, 1, 1 ],
	245	[ 1, 1, 1, 3, 0, 1, 1, 1, 0, 1, 1, 1 ],
	246	[ 1, 1, 1, 0, 3, 1, 1, 1, 0, 1, 1, 1 ],
	247	[ 1, 0, 1, 1, 1, 3, 1, 1, 1, 0, 1, 1 ],
	248	[ 1, 1, 0, 1, 1, 1, 3, 0, 1, 1, 1, 1 ],
	249	[ 1, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1 ],
	250	[ 1, 1, 1, 0, 0, 1, 1, 1, 3, 1, 1, 1 ],
	251	[ 1, 0, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1 ],
	252	[ 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 0 ],
	253	[ 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3 ] ]
	254	gap> InformationMatrix(D);
	255	[ [ 9/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, 0, 0 ],
	256	[ -1/4, 9/4, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4, 0, -1/4, -1/4 ],
	257	[ -1/4, -1/4, 9/4, -1/4, -1/4, -1/4, 0, 0, -1/4, -1/4, -1/4, -1/4 ],
	258	[ -1/4, -1/4, -1/4, 9/4, 0, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4 ],
	259	[ -1/4, -1/4, -1/4, 0, 9/4, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4 ],
	260	[ -1/4, 0, -1/4, -1/4, -1/4, 9/4, -1/4, -1/4, -1/4, 0, -1/4, -1/4 ],
	261	[ -1/4, -1/4, 0, -1/4, -1/4, -1/4, 9/4, 0, -1/4, -1/4, -1/4, -1/4 ],
	262	[ -1/4, -1/4, 0, -1/4, -1/4, -1/4, 0, 9/4, -1/4, -1/4, -1/4, -1/4 ],
	263	[ -1/4, -1/4, -1/4, 0, 0, -1/4, -1/4, -1/4, 9/4, -1/4, -1/4, -1/4 ],
	264	[ -1/4, 0, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4, 9/4, -1/4, -1/4 ],
	265	[ 0, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, 9/4, 0 ],
	266	[ 0, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, 0, 9/4 ] ]
	267	gap> BlockDesignEfficiency(D);
	268	rec( A := 33/41,
	269	CEFpolynomial := x_1^11-9x_1^10+147/4x_1^9-719/8x_1^8+18723/128x_1^7-106\
	270	47/64x_1^6+138159/1024x_1^5-159813/2048x_1^4+2067201/65536x_1^3-556227/655\
	271	36x_1^2+89667/65536x_1-6561/65536, Dpowered := 6561/65536,
	272	Einterval := [ 3/4, 3/4 ] )
	273	gap> BlockDesignEfficiency(D,10^(-4),true);
	274	rec( A := 33/41,
	275	CEFpolynomial := x_1^11-9x_1^10+147/4x_1^9-719/8x_1^8+18723/128x_1^7-106\
	276	47/64x_1^6+138159/1024x_1^5-159813/2048x_1^4+2067201/65536x_1^3-556227/655\
	277	36x_1^2+89667/65536x_1-6561/65536, Dpowered := 6561/65536,
	278	Einterval := [ 3/4, 3/4 ], MV := 3/4 )
	279	gap> D:=PGPointFlatBlockDesign(2,3,1);;
	280	gap> Size(AutGroupBlockDesign(D));
	281	5616
	282	gap> D1:=BlockDesign(3,[[1],[1,2,3],[2]]);;
	283	gap> D2:=BlockDesign(3,[[1],[1,2,3],[3]]);;
	284	gap> IsIsomorphicBlockDesign(D1,D2);
	285	true
	286	gap> D3:=BlockDesign(4,[[1],[1,2,3],[3]]);;
	287	gap> IsIsomorphicBlockDesign(D2,D3);
	288	false
	289	gap> D1:=BlockDesign(3,[[1],[1,2,3],[2]]);;
	290	gap> D2:=BlockDesign(3,[[1],[1,2,3],[3]]);;
	291	gap> D3:=BlockDesign(4,[[1],[1,2,3],[3]]);;
	292	gap> Length(BlockDesignIsomorphismClassRepresentatives([D1,D2,D3]));
	293	2
	294	gap> DL:=BlockDesigns(rec(
	295	> v:=15,blockSizes:=[3],
	296	> tSubsetStructure:=rec(t:=2,lambdas:=[1]),
	297	> requiredAutSubgroup:=
	298	> Group((1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15))));;
	299	gap> List(DL,AllTDesignLambdas);
	300	[ [ 35, 7, 1 ], [ 35, 7, 1 ], [ 35, 7, 1 ] ]
	301	gap> Collected(List(DL,D->Size(AutGroupBlockDesign(D))));
	302	[ [ 5, 1 ], [ 60, 1 ], [ 20160, 1 ] ]
	303	gap> parclasses:=List(DL,D->
	304	> BlockDesigns(rec(
	305	> blockDesign:=D,
	306	> v:=15,blockSizes:=[3],
	307	> tSubsetStructure:=rec(t:=1,lambdas:=[1]))));;
	308	gap> Collected(List(parclasses,Length));
	309	[ [ 1, 2 ], [ 2, 1 ] ]
	310	gap> Collected(List(parclasses,L->Collected(List(L,parclass->Size(parclass.autSubgroup)))));
	311	[ [ [ [ 5, 1 ] ], 1 ], [ [ [ 6, 1 ], [ 60, 1 ] ], 1 ], [ [ [ 360, 1 ] ], 1 ] ]
	312	gap> List([1..6],k->Length(SemiLatinSquareDuals(4,k)));
	313	[ 2, 10, 40, 164, 621, 2298 ]
	314	gap> List([1..6],k->Length(SemiLatinSquareDuals(4,k,"default","default",4)));
	315	[ 2, 11, 46, 201, 829, 3343 ]
	316	gap> DL:=BlockDesigns(rec(
	317	> v:=15,blockSizes:=[3],
	318	> tSubsetStructure:=rec(t:=2,lambdas:=[1]),
	319	> requiredAutSubgroup:=
	320	> Group((1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15))));;
	321	gap> Collected(List(DL,D->Size(AutomorphismGroup(D))));
	322	[ [ 5, 1 ], [ 60, 1 ], [ 20160, 1 ] ]
	323	gap> x:=Indeterminate(Rationals,1);
	324	x_1
	325	gap> f:=(x+3)*(x^2-3);
	326	x_1^3+3x_1^2-3x_1-9
	327	gap> L:=DESIGN_IntervalForLeastRealZero(f,-5,5,10^(-3));
	328	[ -3, -3 ]
	329	gap> L:=DESIGN_IntervalForLeastRealZero(f,-2,5,10^(-3));
	330	[ -14193/8192, -7093/4096 ]
	331	gap> List(L,Float);
	332	[ -1.73254, -1.73169 ]
	333	gap> L:=DESIGN_IntervalForLeastRealZero(f,0,5,10^(-3));
	334	[ 14185/8192, 7095/4096 ]
	335	gap> List(L,Float);
	336	[ 1.73157, 1.73218 ]
	337	gap> L:=DESIGN_IntervalForLeastRealZero(f,0,5,10^(-5));
	338	[ 454045/262144, 908095/524288 ]
	339	gap> List(L,Float);
	340	[ 1.73204, 1.73205 ]
	341	gap> L:=DESIGN_IntervalForLeastRealZero(f,2,5,10^(-5));
	342	[ ]
	343	gap> STOP_TEST( "testall.tst", 10000 );
	344	## The first argument of STOP_TEST should be the name of the test file.
	345	## The number is a proportionality factor that is used to output a
	346	## "GAPstone" speed ranking after the file has been completely processed.
	347	## For the files provided with the distribution this scaling is roughly
	348	## equalized to yield the same numbers as produced by the test file
	349	## tst/combinat.tst. For package tests, you may leave it unchanged.
	350
	351	#############################################################################