[Up] [Previous] [Next] [Index]

# 3 General concepts

### Sections

1. Introduction
2. How partial difference sets are represented
3. Basic functions for startset generation
4. A brute force method

In this chapter, we first give a definition of relative difference sets and outline a part of the theory. Then we have a quick look at the way difference sets are represented in RDS.

After that, some basic methods for the generation of difference sets are explained.

If you already read chapter RDS:A quick start and want to know what `StartsetsInCoset` really does, you may want to read this chapter (and the following one, of course). The main high-level function in this chapter is `ExtendedStartsets`.

## 3.1 Introduction

Inputrds

## 3.2 How partial difference sets are represented

Let G be a group. We define an enumeration {g1,...,gn}=G and represent D ⊆ G as a list of integers (where ,of course, i represents gi for all 1 ≤ i ≤ n). So the automorphism group of G is represented as a permutation group of degree n. One of the operations performed most often by methods in RDS is the calculation of quotients in G. So we calculate a look-up table for this.

This pre-calculation is done by the operation `PermutationRepForDiffsetCalcuations`. So before you start generating difference set, call this function and work with the data structure returned by it.

For an exhaustive search, the ordering of G is very important. To avoid generating duplicate partial difference sets, we would like to represent partial difference sets by sets, i.e. ordered lists. But in fact, RDS does not assume that partial difference sets are sets. The operations `ExtendedStartSets` and `AllDiffsets` assume that the last element of partial difference set is its maximum. But they don't test it. So if you start from scratch, these methods generate difference sets which are really sets. Whereas the `NoSort` versions disregard the ordering of G and will produce duplicates.

The reason for this seemingly strange behaviour is the following: Assume that we have a normal subgroup U ≤ G and know that every difference set D ⊆ G contains exactly ni elements from the ith coset modulo U. Then it is natural to generate difference sets by first searching all partial difference sets of length n1 containing entirely of elements of the first coset modulo U and then proceed with the other cosets.

This method of difference set generation is normally not compatible with the ordering of G. This is why partial difference sets are not required to be sets. See chapter RDS:An Example Program for an example.

## 3.3 Basic functions for startset generation

Defining an enumeration of the a group G, every relative difference set may be represented by a list of integers. Indexing G in this way has the advantage of the automorphism group of G being a permutation group. As relative difference sets are normally calculated in small groups, it is possible to store a complete multiplication table of the group in terms of the enumeration.

If not stated otherwise, partial difference sets are always considered to be lists of integers. Note that it is not required for a partial difference set to be a set.

- `PermutationRepForDiffsetCalculations(` group`) O`
- `PermutationRepForDiffsetCalculations(` group`,` autgrp`) O`

  For a group group, `PermutationRepForDiffsetCalculations(`group`)` returns a record containing:

  1.
    the group .G=group.
  2.
    the sorted list .Glist=`Set(`group`)`,
  3.
    the automorphism group .A of group,
  4.
    the group .Aac, which is the permutation action of A on the indices of .Glist,
  5.
    .Ahom=`ActionHomomorphism(`.A`,`.Glist`)`,
  6.
    the group .Ai of anti-automorphisms of .group acting on the indices of Glist,
  7.
    the multiplication table .diffTable of .group in a special form.

  .diffTable is a matrix of integers defined such that ``.difftable`[i][j]` is the position of ``Glist`[i](`Glist`[j])^-1)` in Glist with ``Glist`[1]=One(`group`)`.

  `PermutationRepForDiffsetCalculations` runs into an error if `Set(`group`)[1]` is not equal to `One(`group`)`.

  If autgrp is given, `PermutationRepForDiffsetCalculations` will not calculate the automorphism group of group but will take autgrp instead without any test.

  If 'Set(group)[1]' is not equal to `One(`group`)`, then `PermutationRepForDiffsetCalculations` returns an error message stating ``Unable to generate Glist''. In this case, calculating a representation helps:

      gap> G:=DirectProduct(SL(2,3),CyclicGroup(3));
      <group of size 72 with 3 generators>
      gap> data:=PermutationRepForDiffsetCalculations(G);
      Error, Unable to generate <Glist>
       called from
      <function>( <arguments> ) called from read-eval-loop
      Entering break read-eval-print loop ...
      you can 'quit;' to quit to outer loop, or
      you can 'return;' to continue
      brk> quit;
      gap> phi:=ActionHomomorphism(G,Set(G),OnRight);
      <action homomorphism>
      gap> Gnew:=ImagesSource(phi);
      <permutation group with 3 generators>
      gap> data:=PermutationRepForDiffsetCalculations(Gnew);

  A partial difference set may be converted from a list of group elements to a list of integers using

- `GroupList2PermList(` list`,` dat`) O`

  where dat is a record containing .diffTable as returned by `PermutationRepForDiffsetCalculations`. The inverse operation is performed by

- `PermList2GroupList(` list`,` dat`) O`

      gap>  G:=DihedralGroup(6);
      <pc group of size 6 with 2 generators>
      gap> N:=NormalSubgroups(G)[2];
      Group([ f2 ])
      gap> dat:=PermutationRepForDiffsetCalculations(G);
      rec( G := <pc group of size 6 with 2 generators>,
        Glist := [ <identity> of ..., f1, f2, f1*f2, f2^2, f1*f2^2 ],
        A := <group of size 6 with 2 generators>,
        Aac := Group([ (3,5)(4,6), (2,4,6) ]),
        Ahom := <action homomorphism>,
        Ai := Group([ (3,5), (3,5)(4,6), (2,4,6) ]),
        diffTable := [ [ 1, 2, 5, 4, 3, 6 ], [ 2, 1, 6, 3, 4, 5 ],
            [ 3, 6, 1, 2, 5, 4 ], [ 4, 5, 2, 1, 6, 3 ],
            [ 5, 4, 3, 6, 1, 2 ], [ 6, 3, 4, 5, 2, 1 ] ] )
      gap> Nperm:=GroupList2PermList(Set(N),dat);
      [ 1, 3, 5 ]

  In the following functions the record dat has to contain a matrix .diffTable as returned by `PermutationRepForDiffsetCalculations`.

- `NewPresentables(` list`,` newel`,` table`) O`
- `NewPresentables(` list`,` newel`,` dat`) O`
- `NewPresentables(` list`,` newlist`,` dat`) O`
- `NewPresentables(` list`,` newlist`,` table`) O`

  `NewPresentables(` list`,`newel`,`dat`)` takes a record dat as returned by `PermutationRepForDiffsetCalculations(`group`)`. For `NewPresentables(` list`,`newel`,`table`)`, table has to be the multiplication table in the form of `NewPresentables(` list`,`newel`,`dat.diffTable`)`

  The method returns the unordered list of quotients d1newel −1 with d1 ∈ list ∪{1} (in permutation representation).

  When used with a list newlist, a list of quotients d1d2−1 with d1 ∈ list ∪{1} and d2 ∈ newlist is returned.

- `AllPresentables(` list`,` table`) O`
- `AllPresentables(` list`,` dat`) O`

  Let list be a list of integers representing elements of a group defined by dat (or table). `AllPresentables(` list`,`table`)` returns an unordered list of quotients ab−1 for all group elements a,b represented by integers in list. If 1 ∈ list , an error is issued. The multiplication table table has to be of the form as returned by `PermutationRepForDiffsetCalculations`. And dat is a record as calculated by `PermutationRepForDiffsetCalculations`.

      gap> G:=CyclicGroup(7);;dat:=PermutationRepForDiffsetCalculations(G);;
      gap> AllPresentables([2,3],dat);
      [ 2, 3, 7, 2, 7, 6 ]
      gap> AllPresentables([1,2,3],dat);
      Error...

- `RemainingCompletions(` diffset`,` completions`[,` forbidden`],` dat`[,` lambda`] ) O`
- `RemainingCompletionsNoSort(` diffset`,` completions`[,` forbidden`],` table`[,` lambda`] ) O`

  For a partial difference set diffset, `RemainingCompletions(`diffset`,`completions`,`dat`)` returns a subset of the set completions, such that each of its elements may be added to diffset without it loosing the property to be a partial difference set. Only elements greater than the last element of diffset are returned.

  For partial relative difference sets, forbidden is the forbidden set.

  `RemainingCompletionsNoSort` does also return elements from completions which are smaller than ``diffset`[Size(`diffset`)]`.

      gap> G:=CyclicGroup(7);
      <pc group of size 7 with 1 generators>
      gap> dat:=PermutationRepForDiffsetCalculations(G);;
      gap> RemainingCompletionsNoSort([4],[1..7],dat);
      [ 2, 3 ]
      gap> RemainingCompletionsNoSort([4],[1..7],dat,2);
      [ 2, 3, 6, 7 ]
      gap> RemainingCompletions([4],[1..7],dat);
      [  ]
      gap> RemainingCompletions([4],[1..7],dat,2);
      [ 6, 7 ]

- `ExtendedStartsets(` startsets`,` completions`, [`forbiddenset`][,` aim`],` Gdata`[,` lambda`] ) O`
- `ExtendedStartsetsNoSort(` startsets`,` completions`, [`forbiddenset`][,` aim`],` Gdata`[,` lambda`] ) O`

  For a set of partial (relative) difference sets startsets, the set of all extensions by one element from completions is returned. Here an ``extension'' of a partial diffence set S is a list which has one element more than S and contains S.

  Here completions is a set of elements wich may be appended to the lists in startsets to generate new partial difference sets. For relative difference sets, the forbidden set forbiddenset must be given. And the integer aim gives the desired total length, i.e. the number of elements of completions that have to be added to each startset plus its length. Note that the elements of startset are always extended by one element (if they can be extended). aim does only tell how many elements from completions you want to add. A partial difference set is only be extended, if there are enough admissible elements in completions, so if for some S ∈ startsets , we have less than aim −`Size`(S) elements in completions which can be added to S, no extension of S is returned.

  If lambda is not passed as a parameter, it is assumed to be 1.

  Note that `ExtendedStartsets` does use `RemainingCompletions` while `ExtendedStartsetsNoSort` uses `RemainingCompletionsNoSort`. Note that the partial difference sets generated with `ExtendedStartsetsNoSort` are not sets (i.e. not sorted). This may result in doing work twice. But it can also be useful, especially when generating difference sets ``coset by coset''.

      gap> G:=CyclicGroup(7);;dat:=PermutationRepForDiffsetCalculations(G);;
      gap> startsets:=[[2],[4],[6]];;
      gap> ExtendedStartsets(startsets,[1..7],dat);
      [ [ 2, 4 ], [ 2, 6 ] ]
      gap> ExtendedStartsets(startsets,[1..7],3,dat);
      [ [ 2, 4 ] ]
      gap> ExtendedStartsets(startsets,[1..7],dat,2);
      [ [ 2, 3 ], [ 2, 4 ], [ 2, 5 ], [ 2, 6 ], [ 4, 6 ], [ 4, 7 ], [ 6, 7 ] ]
      gap> ExtendedStartsetsNoSort(startsets,[1..7],dat);
      [ [ 2, 4 ], [ 2, 6 ], [ 4, 2 ], [ 4, 3 ], [ 6, 2 ], [ 6, 5 ] ]

  ## 3.4 A brute force method

  The following method can be used to find (partial) difference sets by brute force.

- `AllDiffsets(` diffset`,` completions`,` aim`,` forbidden`,` Gdata`,` lambda`) O`

  Let diffset be partial relative difference set and completions a list of possible completions and forbidden the forbidden set. Then `AllDiffsets` returns a list of (partial) difference sets which contain diffset. Gdata is the record as always and lambda is the parameter of the relative difference set. forbidden and completions have to be lists of integers.

  [Up] [Previous] [Next] [Index]