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# 1 Design

### Sections

1. Installing the DESIGN Package
2. Loading DESIGN
3. The structure of a block design in DESIGN
4. Example of the use of DESIGN

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 http://designtheory.org and Dotw. 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 https://www.gnu.org/licenses/gpl.html.

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, https://gap-packages.github.io/design.

For questions, remarks, suggestions, and issues, please use the issue tracker at https://github.com/gap-packages/design/issues.

## 1.1 Installing the DESIGN Package

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 Installing a GAP Package). If you do need to download DESIGN, you can find an archive file for the latest release at https://github.com/gap-packages/design/releases. This archive file can then be downloaded and unpacked in the `pkg` subdirectory of an appropriate GAP root directory (see the main GAP reference section GAP Root Directories).

The DESIGN package is written entirely in GAP code, and requires no further installation. However, DESIGN makes use of the GRAPE package Grape, which must be fully installed.

## 1.2 Loading DESIGN

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

    LoadPackage("design");

## 1.3 The structure of a block design in DESIGN

A block design is an ordered pair (X,B), where X is a non-empty finite set whose elements are called points, and B 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 (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 D is stored as a record, with mandatory components `isBlockDesign`, `v`, and `blocks`. The points of a block design D are always 1,2,...,``D`.v`, but they may also be given names in the optional component `pointNames`, with ``D`.pointNames[`i`]` the name of point i. The `blocks` component must be a sorted list of the blocks of D (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 `isSimple`, `isBinary`, `isConnected`, `r`, `blockSizes`, `blockNumbers`, `resolutions`, `autGroup`, `autSubgroup`, `tSubsetStructure`, `allTDesignLambdas`, `efficiency`, `id`, `statistical_propertiesXML`, and `pointNames`.

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

## 1.4 Example of the use of DESIGN

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,{4,5},1) design D invariant under H=〈(1,2,…,20)〉, and then applies the *-construction of McSo to this design D to obtain a non-simple 2-(21,4,3) design Dstar with automorphism group of order 80. The program then classifies the near-resolutions of Dstar invariant under the subgroup of order 5 of H, and finds exactly two such (up to the action of \Aut(Dstar)). Finally, Dstar is printed.

Further extended examples of the use of the DESIGN package can be found in Soi13 and Soi24.

    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 )

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