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

# 4 Invariants for Difference Sets

### Sections

1. The Coset Signature
2. Blackbox functions

This chapter contains an important tool for the generation of difference sets. It is called the ``coset signature'' and is an invariant for equivalence of partial relative difference sets. For large λ, there is an invariant calculated by `MultiplicityInvariantLargeLambda`. This invariant can be used complementary to the coset signature and is explained in section Ordered signatures by quotient images.

Most of the methods explained here are not commonly used. If you do not want to know how coset signatures work in detail, you can safely skip a large part of this and go straight to the explanation of `SignatureDataForNormalSubgroups` and `ReducedStartsets`.

The last section (RDS:Blackbox functions) of this chapter has some functions which allow the user to use coset signatures with even less effort. But be aware that these functions make choices for you that you probably do not want if you do very involved calculations. In particular, the coset signatures are not stored globally and hence cannot be reused. For a demonstration of these easy-to-use functions, see chapter RDS:A quick start

## 4.1 The Coset Signature

Let R ⊆ G be a (partial) relative difference set (for definition see Introduction) with forbidden set N ⊆ G. Let U ≤ G be a normal subgroup and C={g1,..., g|G:U|} be a system of representatives of G/U.

The intersection number of R with Ugi is defined as vi=|R∩Ugi|. For every normal subgroup U ≤ G the multiset {|R∩Ugi| :gi ∈ C} is called ``coset signature of R (relative to U)''.

Let D ⊆ G be a relative difference set and {v1,...,v|G:U|} its coset signature. Then the following equations hold (see Bruck55,RoederDiss):

| ∑ | vi=k

| ∑ | vi2=λ(|U|−|U∩N|)+k

| ∑ j | vj vij = λ(|U|−|giU ∩N|)

for gi ∉ U

where vij=|D∩gigjU|. If the forbidden set N is a subgroup of G we have |giU∩N| is either 0 or equal to |U∩N|.

Given a group G, the forbidden set N ⊆ G and some normal subgroup U ≤ G, the right sides of this equations are known. So we may ask for tuples (v1,...,v|G:U|) solving this system of equations. Of course, we index the vi with the elements of G/U, so the last equation poses conditions to the ordering of the tuple (v1,...,v|G:U|).

So we call any multiset {v1,...,v|G:U|} solving the above equations an ``admissible signature'' for U.

- `CosetSignatureOfSet(` set`,` cosets`) F`

  `CosetSignatureOfSet(` set`,`cosets`)` returns the ordered list of intersection numbers of set. That is, the size of the intersection of set with each Element of cosets.

  Note that it is not tested, if cosets is really a list of cosets. `CosetSignatureOfSet(` set`,`cosets`)` works for any List set and any list of lists cosets. So be careful!

      gap> G:=SymmetricGroup(5);;
      gap> A:=AlternatingGroup(5);;
      gap> CosetSignatureOfSet([(1,2),(1,5),(1,2,3)],RightCosets(G,A));
      [ 1, 2 ]
      gap> CosetSignatureOfSet([(1,2),(1,5),(1,2,3)],[A]);
      [ 1 ]
      gap> CosetSignatureOfSet([(1,2),(1,5),(1,2,3)],[[(1,2),(1,2,3)],[(3,2,1)]]);
      [ 0, 2 ]

- `CosetSignatures(` Gsize`,` Usize`,` diffsetorder`) O`
- `CosetSignatures(` Gsize`,` Nsize`,` Usize`,` Intersectsizes`,` k`,` lambda`) O`

  `CosetSignatures(` Gsize`,`Usize`,`diffsetorder`)` returns all Gsize /Usize tuples such that the sum of the squares of each tuple equals Usize+diffsetorder. And the sum of each tuple equals diffsetorder+1.

  These are necessary conditions for signatures of difference sets and normal subgroups of order Usize in groups of order Gsize (see The Coset Signature).

  `CosetSignatures(` Gsize`,`Nsize`,`Usize`,`Intersectsizes`,`k`,`lambda`)` Calculates all multiset meeting some conditions for signatures of relative difference sets and normal subgroups of order Usize in groups of order Gsize (see The Coset Signature). Here Nsize is the size of the forbidden group, Intersectsizes is a list of integers determining the size of the intersection of the forbidden set and the normal Subgroup of order Usize. The pararmeters k and lambda are the usual ones for designs. `CosetSignatures` returns a list containing one pair for each entry i of Intersectsizes. The first entry of this pair is [Gsize ,Nsize ,Usize ,i ,k ,lambda ] and the second one is a list of admissible signatures with these parameters.

      gap> CosetSignatures(256,16,64,[1,4,8,16],17,1);
      [ [ [ 256, 16, 64, 1, 17, 1 ], [  ] ],
        [ [ 256, 16, 64, 4, 17, 1 ], [ [ 3, 4, 4, 6 ] ] ],
        [ [ 256, 16, 64, 8, 17, 1 ], [ [ 4, 4, 4, 5 ] ] ],
        [ [ 256, 16, 64, 16, 17, 1 ], [  ] ] ]
      #And for an ordinary difference set of order 16.
      gap> CosetSignatures(273,1,39,[1],17,1);
      [ [ [ 273, 1, 39, 1, 17, 1 ],
        [ [ 0, 1, 2, 3, 3, 4, 4 ], [ 0, 2, 2, 2, 3, 3, 5 ],
          [ 1, 1, 1, 2, 4, 4, 4 ], [ 1, 1, 1, 3, 3, 3, 5 ],
          [ 1, 1, 2, 2, 2, 4, 5 ] ] ] ]

- `TestSignatureLargeIndex(` sig`,` group`,` Normalsg`[,` factorgrp`] ) O`

  this does only work for ordinary difference sets, not for relative difference sets in general

  `TestSignatureLargeIndex(`sig`,`group`,`Normalsg`[,`factorgrp`])` tests if sig meets some necessary conditions of The Coset Signature to be a signature for a difference set in group for the normal subgroup Normalsg. factorgrp is the factorgroup group/Normalsg. The returned value is true or false resp.

- `TestSignatureCyclicFactorGroup(` sig`,` Nsize`) O`

  This does only work for ordinary difference sets, not for relative difference sets in general

  `TestSignatureCyclicFactorGroup(`sig`,`Nsize`)` test if sig meets meets some necessary conditions of The Coset Signature to be a signature for a difference set in some group, which has a normal subgroup of size Nsize such that the factor group is cyclic. The returned value is true or false resp.

- `TestedSignatures(` sigs`,` group`,` normalsg`[,` maxtest`][,` moretest`] ) O`

  this does only work for ordinary difference sets, not for relative difference sets in general

  Let sigs be a list of possible signatures as returned from `CosetSignatures`. Let normalsg be a subgroup of group. For each signature in sigs, the necessary conditions described in The Coset Signature are tested to decide if the signature can be a signature of a difference set in group for for the normal subgroup normalsg.

  As this involves computation for all permutations of the signature, this can be very costly. The argument maxtest determines how many permutations are admissible. If maxtest=0, all signatures are tested, regardless of how much work is necessary for this. If a signature has too many permutations, it is returned without test. Even though it is not wise, ``maxtest`=0` is the default option. If `InfoLevel(InfoRDS)`indexInfoRDS@ttInfoRDS is at least 2, information about skipped signatures is echoed.

  If the boolean value moretest is false and all signatures in sigs but the last one are found to be not admissible, the last one is returned without test. This saves the time to test the last signature, but if chances are that there is no difference set in group, this may also give away a chance to find out early (every difference set has signatures, so no admissible signature means that no difference set can exist). Default is true.

  `TestedSignatures` calls `TestSignatureCyclicFactorGroup` or `TestSignatureLargeIndex` and returns a sublist of sigs.

      G:=SmallGroup(273,2);
      gap> N:=First(NormalSubgroups(G),g->Order(g)=39);
      Group([ f1, f3 ])
      gap> sigs:=CosetSignatures(273,1,39,[1],17,1);
      [ [ [ 273, 1, 39, 1, 17, 1 ],
        [ [ 0, 1, 2, 3, 3, 4, 4 ], [ 0, 2, 2, 2, 3, 3, 5 ],
          [ 1, 1, 1, 2, 4, 4, 4 ], [ 1, 1, 1, 3, 3, 3, 5 ],
          [ 1, 1, 2, 2, 2, 4, 5 ] ] ] ]
      gap> TestedSignatures(sigs[1][2],G,N);
      [ [ 1, 1, 1, 2, 4, 4, 4 ], [ 1, 1, 1, 3, 3, 3, 5 ] ]

- `TestedSignaturesRelative(` sigs`,` fgdata`, [,` maxtest`][,` moretest`] ) O`

  `TestedSignaturesRelative` takes a list sigs of lists of integers and returns a those which may be signatures of relative difference sets with forbidden set.

  fgdata is a record as returned by `RDSFactorGroupData(`U`,`N`,`lambda`,`Gdata`)` If maxtest is set, a signature s is only tested if `NrPermutationsList(s)` is less than maxtest if maxtest is set to 0, all signatures are tested this is the default. If moretest is tue, a signature is tested even if it is the only one left. This means we do not assume that there must be an admissable signature at all. The default for moretest is true.

- `SigInvariant(` prd `,` data`) O`

  Given a partial relative difference set prd and a list of records with entries cosets and sigs. Here cosets is a full list of cosets and sigs is a list of signatures that may occur for relative difference sets.

  For each record rec in data, the intersection numbers of prd with the cosets of rec.cosets are computed stored in a set sig. If none of the signatures in rec.sigs is pointwise greater or equal sig, `SigInvariant(` prd`,`data`) returns` fail'. Otherwise sig is added to a list of signatures that is returned.

  Note the returned invariant is that of prd∪{1}. The output from `SignatureDataForNormalSubgroups` can be used as data.

- `RDSFactorGroupData(` U`,` N`,` lambda`,` Gdata`) O`

  takes the subgroup U of G, the forbidden set N as a subgroup or subset of G and the record of data Gdata as returned by `PermutationRepForDiffsetCalculations(`G`)` and returns a record containing

  .fg
    the factor group modulo U
  .fglist
    the factor group as a strictly ordered list
  .cosets
    the cosets modulo U as lists of integers
  .lambda
    the parameter lambda as passed to the function
  .Usize
    the size of U
  .fgaut
    the automorphism group of .fg
  .Nfg
    the image of N in .fg
  .fgintersect
    a list of pairs such that the ith entry is the pair consisting of .fg[i] and the size of the intersection of .fg with .Nfg as cosets modulo U.
  .intersectshort
    ist just the second component of .fgintersect.

- `MatchingFGDataNonGrp(` fgdatalist`,` fgmatchdata`) O`

  Let fgdatalist be a list of records and fgmatchdata a record with components .fg, .Nfg and .fgintersect as returned by `RDSFactorGroupData`. Then `MatchingFGDataNonGrp` returns the entry of fgdatalist that defines the same admissible signatures as fgmatchdata. If no such entry exists, `fail` is returned.

  The forbidden set N is not assumed to be a group.

- `MatchingFGData(` fgdatalist`,` fgmatchdata`) O`

  Let fgdatalist be a list of records and fgmatchdata a record with components .fg, .Nfg, .fgintersect and .fgaut as returned by `RDSFactorGroupData`. Then `MatchingFGDataNonGrp` returns the entry of fgdatalist that defines the same admissible signatures as fgmatchdata. If no such entry exists, `fail` is returned.

  Here the forbidden set N has to be a group.

- `SignatureDataForNormalSubgroups(` Normals`,` globalSigData`,` forbiddenSet`,` Gdata`,` parameters`) O`

  Let Gdata be a record as returned by `PermutationRepForDiffsetCalculations`. Let Normals be a list of normal subgroups of Gdata.G, and forbiddenSet the forbidden set (as set of group elements or group).

  parameters must be a list of length 4 of the form [k,lambda,maxtest,moretest] with k the length of the relative difference set to be constructed and lambda the parameter as always. maxtest and moretest are passed to `TestedSignaturesRelative` and must be set.

  `SignatureDataForNormalSubgroups` returns a list containing one record for each group U in Normals. This record contains:

  1.
    the subgroup U named .subgroup
  2.
    the signatures .sigs for U
  3.
    the cosets .cosets modulo U as lists of integers

  Moreover, the list globalSigData is used to store global information which can be reused with other groups. The ith entry of globalSigData is a list of records that contains all known information about subgroups of order i. Each of these records has the following components:

  1.
    .cspara the parameters for `CosetSignatures`
  2.
    .sigs the output of `CosetSignatures` when the input is .cspara
  3.
    .fgsigs a list of records containing data about factor groups with parameters .cspara:
    3.1.
      .fg the factor group
    3.2.
      .fgaut the automorphism group of .fg
    3.3.
      .Nfg the image of the forbidden set N under the natural epimorphism to .fg
    3.4.
      .fgintersect the pairs [g,|g∩N| ] for all g in .fg. Here N is the forbidden set.
    3.5.
      .sigs the known admissible signatures (this is a subset of the set in number 2. of course)

    The list globalSigData can be used if different groups are studied. If a group has a normal subgroup with parameters (in the sense of .cspara) listed in globalSigData, the signatures from a previous calculation may be used. Of course, the factor groups have to be checked first. This check is done with `MatchingFGData` or `MatchingFGDataNonGrp`.

    So the second run of `SignatureDataForNormalSubgroups` with the same parameters and different Gdata and Normals will normally be much faster, as the signatures are already stored in globalSigData. Note that maxtest and moretest are not stored. So a second run with larger maxtest will not result in a recalculation of signatures.

- `ReducedStartsets(` startsets`,` autlist`,` csdata`,` Gdata`) O`
- `ReducedStartsets(` startsets`,` autlist`,` func`,` Gdata`) O`

  Let startsets be a set of partial relative difference sets, autlist a list of permutation groups and Gdata record returned by `PermutationRepForDiffsetCalculations`. Then `ReducedStartsets` partitions the list startsets according to the values of the function func and performs a test for equivalence on the elements of the partition. The list returned is a sublist of startsets of pairwise non-equivalent partial relative difference sets if func is an invariant for partial relative difference sets. All elements for which func returns `fail` are discarded.

  Let csdata be a list of records as used for `SigInvariant` (i.e. containing .cosets and .signatures). Then `ReducedStartsets(`startsets`,`autlist`,`csdata`,`Gdata`)` `SigInvariant` is used for func.

- `maxAutsizeForOrbitCalculation V`

  In `ReducedStartsets`, a bound is needed to decide if `Orbit` or `RepresentativeAction` should be used. If the group is larger than maxAutsizeForOrbitCalculation, `RepresentativeAction` is used. The default value for `maxAutsizeForOrbitCalculation` is 106. If you want to change it, you will have to edit the file `sigs.gd`.

  ## 4.2 Blackbox functions

  Here are a few functions used in chapter RDS:A quick start. These are meant as black boxes for quick tests. Some of them make choices for you which might not be suitable to the chase you consider, so for serious studies, consider using the more complicated-looking functions above (an example for this comprises chapter RDS:An Example Program).

- `SignatureData(` Gdata`,` forbiddenSet`,` k`,` lambda`,` maxtest`) O`

  Let Gdata be a record as returned by `PermutationRepForDiffsetCalculations`. Let forbiddenSet the forbidden set (as set or group).

  k is the length of the relative difference set to be constructed and lambda the usual parameter. maxtest is the Then `SignatureData` calls `SignatureDataForNormalSubgroups` for normal subgroups of order at least `RootInt(Gdata.G)`. Here maxtest is an integer which determines how many permutations of a possible signature are checked to be a sorted signature. Choose a value of at least 105. Larger numbers here normaly result in better results when generating difference sets (making reduction more effective).

- `NormalSgsHavingAtMostNSigs(` sigdata`,` n`,` lengthlist`) F`

  Let sigdata be a list as returned by 'SignatureDataForNormalSubgroups', an integer n and a list of integers lengthlist. `NormalSgsHavingAtMostKSigs` filters sigdata and returns a list of records with components .subgroup and .sigs is returned, such that for every entry .subgroup is a normal subgroup of index in lengthlist having at most n signatures.

- `SuitableAutomorphismsForReduction(` Gdata`,` normalsg`) F`

  Given a normal subgroup normalsg of Gdata.G, the function returns a list containing the group of automorphisms of Gdata.G which stabilizes all cosets modulo normalsg. This group is returned as a group of permutations on Gdata.Glist (which is actually the right regular representation). The returned list can be used with `StartsetsInCoset`.

- `StartsetsInCoset(` ssets`,` coset`,` forbiddenSet`,` aim`,` autlist`,` sigdat`,` data`,` lambda`) F`

  Assume, we want to generate difference sets ``coset by coset'' modulo some normal subgroup. Let ssets be a (possibly empty) set of startsets, coset the coset from which to take the elements to append to the startsets from ssets. Furthermore, let aim be the size of the generated partial difference sets (that is, the size of the elements from ssets plus the number of elements to be added from coset). Let autlist be a list of groups of automorphisms (in permutation representation) to use with the reduction algorithm. Here the output from `SuitableAutomorphismsForReduction` can be used. And data and sigdat are the records as returned by `PermutationRepForDiffsetCalculations` and `SignatureDataForNormalSubgroups` (or `SignatureData`, alternatively). The parameter lambda is the usual one for difference sets (the number of ways of expressing elements outside the forbidden set as quotients).

  Then `StartsetsInCoset` returns a list of partial difference sets (a list of lists of integers) of length aim.

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