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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 1.6 package for GAP (version at least 4.5). The DESIGN package is for constructing, classifying, partitioning and studying block designs.

All DESIGN functions are written entirely in the GAP language. However, DESIGN requires the GRAPE package Grape (version at least 4.4) to be installed, and makes use of certain GRAPE functions, some of which make use of B.D. McKay's nauty package Nauty. These GRAPE functions can only be used on a fully installed version of GRAPE.

The DESIGN package is Copyright © Leonard H. Soicher 2003--2011. 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.

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 http://www.gnu.org/licenses/gpl.html

If you use DESIGN to solve a problem then please send a short email about it to L.H.Soicher@qmul.ac.uk, and reference the DESIGN package as follows:

L.H. Soicher, The DESIGN package for GAP, Version 1.6, 2011, http://designtheory.org/software/gap_design/.

Any comments or bug reports should also go to L.H.Soicher@qmul.ac.uk.

## 1.1 Installing the DESIGN Package

The DESIGN package is included in the standard GAP 4.5 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 Reference: Installing a GAP Package). If you do need to download DESIGN, you can find archive files for the package in various formats at http://www.gap-system.org/Packages/design.html, and then your archive file of choice should be downloaded and unpacked in the `pkg` subdirectory of an appropriate GAP root directory (see the main GAP reference section Reference: GAP Root Directory).

The DESIGN package is written entirely in GAP code, and requires no further installation. However, the DESIGN package has complete functionality only in an environment in which the GRAPE package is fully installed. Usually this will be on a Unix system.

## 1.2 Loading DESIGN

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

    gap> LoadPackage("design");
    true

## 1.3 The structure of a block design in DESIGN

A block designindexblock 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 designsindexbinary block design (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=langle (1,2,...,20)rangle, 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.

    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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