- Installing the DESIGN Package
- Loading DESIGN
- The structure of a block design in DESIGN
- Example of the use of DESIGN

This manual describes the DESIGN 1.7 package for GAP. The DESIGN package is for constructing, classifying, partitioning, and studying block designs.

The DESIGN package is Copyright © Leonard H. Soicher 2003--2019. 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

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

L.H. Soicher, The DESIGN package for GAP, Version 1.7, 2019, https://gap-packages.github.io/design.

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

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 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 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.

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

LoadPackage("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.

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.

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 )

design manual

March 2019