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<h1>1 Design</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP001.htm#SECT001">Installing the DESIGN Package</a>
<li> <A HREF="CHAP001.htm#SECT002">Loading DESIGN</a>
<li> <A HREF="CHAP001.htm#SECT003">The structure of a block design in DESIGN</a>
<li> <A HREF="CHAP001.htm#SECT004">Example of the use of DESIGN</a>
</ol>

This manual describes the DESIGN 1.8.2 package for GAP.
The DESIGN package is for constructing, classifying, partitioning,
and studying block designs.

The DESIGN package is Copyright © Leonard H. Soicher
2003--2024. DESIGN is part of a wider project, which received EPSRC
funding under grant GR/R29659/01, to provide a web-based resource for
design theory; see <a href="http://designtheory.org">http://designtheory.org</a> and <a href="biblio.htm#Dotw"><cite>Dotw</cite></a>.
The development of DESIGN was also partially supported by EPSRC grant
EP/M022641/1 (CoDiMa: a Collaborative Computational Project in the area
of Computational Discrete Mathematics).

DESIGN is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version. For details, see
<a href="https://www.gnu.org/licenses/gpl.html">https://www.gnu.org/licenses/gpl.html</a>.

Please reference your use of the DESIGN package in a published work
as follows:

L.H. Soicher, The DESIGN package for GAP, Version 1.8.2, 2024,
<a href="https://gap-packages.github.io/design">https://gap-packages.github.io/design</a>.

For questions, remarks, suggestions, and issues, please use the issue
tracker at <a href="https://github.com/gap-packages/design/issues">https://github.com/gap-packages/design/issues</a>.


<h2><a name="SECT001">1.1 Installing the DESIGN Package</a></h2>

The DESIGN package is included in the standard GAP distribution. You
only need to download and install DESIGN if you need to install
the package locally or are installing an upgrade of DESIGN to
an existing installation of GAP (see the main GAP reference
section <a href="../../../doc/ref/chap76.html#X82473E4B8756C6CD">Installing a GAP Package</a>). If you do need to download
DESIGN, you can find an archive file for the latest release at
<a href="https://github.com/gap-packages/design/releases">https://github.com/gap-packages/design/releases</a>. This archive
file can then be downloaded and unpacked in the <code>pkg</code> subdirectory of
an appropriate GAP root directory (see the main GAP reference section
<a href="../../../doc/ref/chap9.html#X7A4973627A5DB27D">GAP Root Directories</a>).

The DESIGN package is written entirely in GAP code, and requires
no further installation. However, DESIGN makes use of the GRAPE
package <a href="biblio.htm#Grape"><cite>Grape</cite></a>, which must be fully installed.


<h2><a name="SECT002">1.2 Loading DESIGN</a></h2>

Before using DESIGN you must load the package within GAP by calling

<pre>
LoadPackage("design");
</pre>


<h2><a name="SECT003">1.3 The structure of a block design in DESIGN</a></h2>

A block design
<a name = "I0"></a>

is an ordered pair <var>(X,B)</var>, where
<var>X</var> is a non-empty finite set whose elements are called points, and
<var>B</var> is a non-empty finite multiset whose elements are called blocks,
such that each block is a non-empty finite multiset of points.

DESIGN deals with arbitrary block designs. However, at present, some
DESIGN functions only work for binary block designs
<a name = "I1"></a>

(i.e. those with no repeated element in any block of
the design), but these functions will check if an input block design
is binary.

In DESIGN, a block design <var>D</var> is stored as a record, with mandatory
components <code>isBlockDesign</code>, <code>v</code>, and <code>blocks</code>. The points of a block
design <var>D</var> are always 1,2,...,<code></code><var>D</var><code>.v</code>, but they may also be given names
in the optional component <code>pointNames</code>, with <code></code><var>D</var><code>.pointNames[</code><var>i</var><code>]</code>
the name of point <var>i</var>. The <code>blocks</code> component must be a sorted list
of the blocks of <var>D</var> (including any repeats), with each block being a
sorted list of points (including any repeats).

A block design record may also have some optional components which store
information about the design. At present these optional components include
<code>isSimple</code>, <code>isBinary</code>, <code>isConnected</code>, <code>r</code>, <code>blockSizes</code>, <code>blockNumbers</code>,
<code>resolutions</code>, <code>autGroup</code>, <code>autSubgroup</code>, <code>tSubsetStructure</code>,
<code>allTDesignLambdas</code>, <code>efficiency</code>, <code>id</code>, <code>statistical_propertiesXML</code>,
and <code>pointNames</code>.

A non-expert user should only use functions in the DESIGN package to
create block design records and their components.


<h2><a name="SECT004">1.4 Example of the use of DESIGN</a></h2>

To give you an idea of the capabilities of this package, we now give
an extended example of an application of the DESIGN package, in
which a nearly resolvable non-simple 2-(21,4,3) design is constructed
(for Donald Preece) via a pairwise-balanced design. All the DESIGN
functions used here are described in this manual.

The program first discovers the unique (up to isomorphism)
pairwise-balanced 2-(21,<var>{4,5}</var>,1) design <var>D</var> invariant under <var>H=langle
(1,2,...,20)rangle</var>, and then applies the <var>*</var>-construction of
<a href="biblio.htm#McSo"><cite>McSo</cite></a> to this design <var>D</var> to obtain a non-simple 2-(21,4,3) design
<var>Dstar</var> with automorphism group of order 80. The program then classifies
the near-resolutions of <var>Dstar</var> invariant under the subgroup of order 5
of <var>H</var>, and finds exactly two such (up to the action of <var>Aut(Dstar)</var>).
Finally, <var>Dstar</var> is printed.

Further extended examples of the use of the DESIGN package can be
found in <a href="biblio.htm#Soi13"><cite>Soi13</cite></a> and <a href="biblio.htm#Soi24"><cite>Soi24</cite></a>.

<pre>
gap> H:=CyclicGroup(IsPermGroup,20);
Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20) ])
gap> D:=BlockDesigns(rec(v:=21,blockSizes:=[4,5],
> tSubsetStructure:=rec(t:=2,lambdas:=[1]),
> requiredAutSubgroup:=H ));;
gap> Length(D);
1
gap> D:=D[1];;
gap> BlockSizes(D);
[ 4, 5 ]
gap> BlockNumbers(D);
[ 20, 9 ]
gap> Size(AutGroupBlockDesign(D));
80
gap> Dstar:=TDesignFromTBD(D,2,4);;
gap> AllTDesignLambdas(Dstar);
[ 105, 20, 3 ]
gap> IsSimpleBlockDesign(Dstar);
false
gap> Size(AutGroupBlockDesign(Dstar));
80
gap> near_resolutions:=PartitionsIntoBlockDesigns(rec(
> blockDesign:=Dstar,
> v:=21,blockSizes:=[4],
> tSubsetStructure:=rec(t:=0,lambdas:=[5]),
> blockIntersectionNumbers:=[[ [0] ]],
> requiredAutSubgroup:=SylowSubgroup(H,5) ));;
gap> Length(near_resolutions);
2
gap> List(near_resolutions,x->Size(x.autGroup));
[ 5, 20 ]
gap> Print(Dstar,"\n");
rec(
 isBlockDesign := true,
 v := 21,
 blocks := [ [ 1, 2, 4, 15 ], [ 1, 2, 4, 15 ], [ 1, 2, 4, 15 ],
 [ 1, 3, 14, 20 ], [ 1, 3, 14, 20 ], [ 1, 3, 14, 20 ], [ 1, 5, 9, 13 ],
 [ 1, 5, 9, 17 ], [ 1, 5, 13, 17 ], [ 1, 6, 11, 16 ], [ 1, 6, 11, 21 ],
 [ 1, 6, 16, 21 ], [ 1, 7, 8, 10 ], [ 1, 7, 8, 10 ], [ 1, 7, 8, 10 ],
 [ 1, 9, 13, 17 ], [ 1, 11, 16, 21 ], [ 1, 12, 18, 19 ],
 [ 1, 12, 18, 19 ], [ 1, 12, 18, 19 ], [ 2, 3, 5, 16 ], [ 2, 3, 5, 16 ],
 [ 2, 3, 5, 16 ], [ 2, 6, 10, 14 ], [ 2, 6, 10, 18 ], [ 2, 6, 14, 18 ],
 [ 2, 7, 12, 17 ], [ 2, 7, 12, 21 ], [ 2, 7, 17, 21 ], [ 2, 8, 9, 11 ],
 [ 2, 8, 9, 11 ], [ 2, 8, 9, 11 ], [ 2, 10, 14, 18 ], [ 2, 12, 17, 21 ],
 [ 2, 13, 19, 20 ], [ 2, 13, 19, 20 ], [ 2, 13, 19, 20 ],
 [ 3, 4, 6, 17 ], [ 3, 4, 6, 17 ], [ 3, 4, 6, 17 ], [ 3, 7, 11, 15 ],
 [ 3, 7, 11, 19 ], [ 3, 7, 15, 19 ], [ 3, 8, 13, 18 ], [ 3, 8, 13, 21 ],
 [ 3, 8, 18, 21 ], [ 3, 9, 10, 12 ], [ 3, 9, 10, 12 ], [ 3, 9, 10, 12 ],
 [ 3, 11, 15, 19 ], [ 3, 13, 18, 21 ], [ 4, 5, 7, 18 ], [ 4, 5, 7, 18 ],
 [ 4, 5, 7, 18 ], [ 4, 8, 12, 16 ], [ 4, 8, 12, 20 ], [ 4, 8, 16, 20 ],
 [ 4, 9, 14, 19 ], [ 4, 9, 14, 21 ], [ 4, 9, 19, 21 ], [ 4, 10, 11, 13 ],
 [ 4, 10, 11, 13 ], [ 4, 10, 11, 13 ], [ 4, 12, 16, 20 ],
 [ 4, 14, 19, 21 ], [ 5, 6, 8, 19 ], [ 5, 6, 8, 19 ], [ 5, 6, 8, 19 ],
 [ 5, 9, 13, 17 ], [ 5, 10, 15, 20 ], [ 5, 10, 15, 21 ],
 [ 5, 10, 20, 21 ], [ 5, 11, 12, 14 ], [ 5, 11, 12, 14 ],
 [ 5, 11, 12, 14 ], [ 5, 15, 20, 21 ], [ 6, 7, 9, 20 ], [ 6, 7, 9, 20 ],
 [ 6, 7, 9, 20 ], [ 6, 10, 14, 18 ], [ 6, 11, 16, 21 ],
 [ 6, 12, 13, 15 ], [ 6, 12, 13, 15 ], [ 6, 12, 13, 15 ],
 [ 7, 11, 15, 19 ], [ 7, 12, 17, 21 ], [ 7, 13, 14, 16 ],
 [ 7, 13, 14, 16 ], [ 7, 13, 14, 16 ], [ 8, 12, 16, 20 ],
 [ 8, 13, 18, 21 ], [ 8, 14, 15, 17 ], [ 8, 14, 15, 17 ],
 [ 8, 14, 15, 17 ], [ 9, 14, 19, 21 ], [ 9, 15, 16, 18 ],
 [ 9, 15, 16, 18 ], [ 9, 15, 16, 18 ], [ 10, 15, 20, 21 ],
 [ 10, 16, 17, 19 ], [ 10, 16, 17, 19 ], [ 10, 16, 17, 19 ],
 [ 11, 17, 18, 20 ], [ 11, 17, 18, 20 ], [ 11, 17, 18, 20 ] ],
 autGroup := Group( [ ( 2,14,10,18)( 3, 7,19,15)( 4,20, 8,12)( 5,13,17, 9),
 ( 1,17, 5, 9)( 2,10,14, 6)( 4,16,12,20)( 7,15,19,11),
 ( 1,18,19,12)( 2,11, 8, 9)( 3, 4,17, 6)( 5,10,15,20)( 7,16,13,14) ] ),
 blockSizes := [ 4 ],
 isBinary := true,
 allTDesignLambdas := [ 105, 20, 3 ],
 isSimple := false )
</pre>

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