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 `lambda`, there is an invariant calculated by
MultiplicityInvariantLargeLambda. This invariant can be used
complementary to the coset signature and is explained in section
RDS:An invariant for large lambda.

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 functions RDSFactorGroupData, MatchingFGData will be interesting for you, if you look for difference sets with the same parameters in different gorups. SignatureDataForNormalSubgroups and SigInvariant

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 basic example

Let `R subseteqG` be a (partial) relative difference set (for
definition see Introduction) with forbidden set `NsubseteqG`. Let
`UleqG` be a normal subgroup and `C={g _{1},..., g_{|G:U|}}` be a
system of representatives of

The intersection number of `R` with `Ug _{i}` is defined as

Let `DsubseteqG` be a relative difference set and
`{v _{1},...,v_{|G:U|}}` its coset signature. Then the following
equations hold (see Bruck55,RoederDiss):

`
matrix
sumv _{i}=k
sumv_{i}^{2}=lambda(|U|-|UcapN|)+k
sum_{j} v_{j} v_{ij}=
lambda(|U|-|g_{i}U capN|)for g_{i}notinU
`

where `v _{ij}=|Dcapg_{i}g_{j}U|`. If the forbidden set

Given a group `G`, the forbidden set `NsubseteqG` and some normal
subgroup `UleqG`, the right sides of this equations are known. So we
may ask for tuples `(v _{1},...,v_{|G:U|})` solving this system of
equations. Of course, we index the

So we call any multiset `{v _{1},...,v_{|G:U|}}` solving the above
equations an ``admissible signature'' for

`CosetSignatureOfSet( `

`, `

` ) 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( `

`, `

`, `

` ) O`

`CosetSignatures( `

`, `

`, `

`, `

`, `

`, `

` ) O`

`CosetSignatures( `

`Gsize``,`

`Usize``,`

`diffsetorder``)`

returns all
` Gsize/Usize` tuples such that the sum of the squares of each tuple
equals

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( `

`, `

`, `

`[, `

`] ) 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( `

`, `

` ) 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( `

`, `

`, `

`[, `

`][, `

`] ) 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 by
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`.

gap> 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( `

`, `

`, [, `

`][, `

`] ) 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( `

`, `

` ) O`

Given a partial relative difference set `diffset` 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
`diffset` 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( `

`diffset``,`

`data``) returns `fail`

.
Otherwise `sig` is added to a list of signatures that is returned.

Note the returned invariant is that of `diffsetcup{1}`.
The output from `SignatureDataForNormalSubgroups`

can be used as `data`.

gap> G:=SmallGroup(273,2); <pc group of size 273 with 3 generators> gap> Gdata:=PermutationRepForDiffsetCalculations(G);; 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 ] ] gap> sigs:=TestedSignatures(sigs[1][2],G,N); [ [ 1, 1, 1, 2, 4, 4, 4 ], [ 1, 1, 1, 3, 3, 3, 5 ] ] gap> ## calculate cosets in permutation notation: gap> rc:=List(RightCosets(G,N),i->GroupList2PermList(Set(i),Gdata));; gap> data:=[rec(cosets:=rc,sigs:=sigs)];; gap> SigInvariant([3,4,5],data); [ [ [ 0, 0, 0, 0, 0, 1, 3 ], 1 ] ]

For an example using SignatureDataForNormalSubgroups see the example after RDS:ReducedStartsets below.

`SignatureDataForNormalSubgroups( `

`, `

`, `

`, `

`, `

` ) 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 `i ^{th}` entry of

- 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,|gcapN| ]`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.gap> G:=CyclicGroup(57); <pc group of size 57 with 2 generators> gap> Gdata:=PermutationRepForDiffsetCalculations(G);; gap> SignatureDataForNormalSubgroups(NormalSubgroups(Gdata.G),sigdata, > [One(Gdata.G)],Gdata,[8,1,10^6,true]); # for ordinary diffset of order 7. [ rec( subgroup := Group([ f1*f2^6 ]), sigs := [ [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2 ] ], cosets := [ [ 1, 20, 40 ], [ 3, 23, 43 ], [ 6, 26, 46 ], [ 9, 29, 49 ], [ 12, 32, 52 ], [ 15, 35, 55 ], [ 18, 38, 57 ], [ 4, 21, 41 ], [ 7, 24, 44 ], [ 10, 27, 47 ], [ 13, 30, 50 ], [ 16, 33, 53 ], [ 19, 36, 56 ], [ 2, 22, 39 ], [ 5, 25, 42 ], [ 8, 28, 45 ], [ 11, 31, 48 ], [ 14, 34, 51 ], [ 17, 37, 54 ] ] ), rec( subgroup := Group([ f2 ]), sigs := [ [ 1, 3, 4 ] ], cosets := [ [ 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54 ], [ 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56 ], [ 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 57 ] ] ) ] gap> Filtered([1..Size(sigdata)],i->IsBound(sigdata[i])); [ 3, 19 ] gap> Size(sigdata[3]); 2 gap> sigdata[3][1].cspara;sigdata[3][2].cspara; [ 57, 1, 3, 1, 7, 1 ] [ 57, 1, 3, 1, 8, 1 ]

The following three functions are used by SignatureDataForNormalSubgroups. If you do not want to write your own function for signature management, you might not need them.

`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
`i`entry is the pair consisting of^{th}`.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.`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.If a list

`csdata`of records as used for SigInvariant (i.e. containing`.cosets`and`.signatures`) is pased, then`ReducedStartsets`

uses SigInvariant for`func`.gap> G:=CyclicGroup(57); <pc group of size 57 with 2 generators> gap> Gdata:=PermutationRepForDiffsetCalculations(G);; gap> cosetsigs:=SignatureDataForNormalSubgroups(NormalSubgroups(Gdata.G), > sigdata, [One(Gdata.G)],Gdata,[8,1,10^6,true]);; gap> SigInvariant([3,4,5,9],cosetsigs); [ [ [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1 ], 1 ], [ [ 1, 1, 3 ], 1 ] ] gap> ssets:=AllDiffsets([],2,[],Gdata);; gap> Size(ssets); 1458 gap> Size(ReducedStartsets(ssets,[Group(())],cosetsigs,Gdata)); #I Size 1458 #I 5/ 0 @ 0:00:00.126 486 gap> Size(ReducedStartsets(ssets,[Gdata.Ai],cosetsigs,Gdata)); #I Size 1458 #I 5/ 0 @ 0:00:00.123 17

`MaxAutsizeForOrbitCalculation V`

In ReducedStartsets, a bound is needed to decide if

`Orbit`

or`RepresentativeAction`

should be used. If the group is larger than`MaxAutsizeForOrbitCalculation@RDS`,`RepresentativeAction`

is used. The default value for`maxAutsizeForOrbitCalculation`

is`5*10`. If you want to change it, you will have to edit the file^{6}`sigs.gd`

.## 5.2 An invariant for large lambda

`MultiplicityInvariantLargeLambda(`

`set``,`

`Gdata``) O`

Let

`set`be a partial relative difference set with`lambda>1`. Set`P``:=AllPresentables(`

`set``,`

`Gdata``)`

then the set of multiplicities of`P`is an invariant for partial relative difference sets.`MultiplicityInvariantLargeLambda`

returns a list in a form as`Collected`

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

(Read this output as: two elements occur once and two occur twice).This invariant can be used for ReducedStartsets complementary to the signature invariant by defining

gap> partfunc:=function(list) > local sig; > if sig=fail > then return fail; > fi; > return [MultiplicityInvariantLargeLambda(list,Gdata),SigInvariant(list,sigdata)]; > end; function( list ) ... end

`partfunc`can then be passed to ReducedStartsets. Of course,`sigdata`has to be the list of records defining the coset signatures.## 5.3 Blackbox functions

Here are a few functions used in chapter RDS:A basic example. 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``) F`

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`10`. Larger numbers here normaly result in better results when generating difference sets (making reduction more effective).^{5}`SigntureData`

chooses normal subgroups of`Gdata.G`and uses SignatureDataForNormalSubgroups to calculate signature data. The global data generated by SignatureDataForNormalSubgroups is just discarded.gap> G:=CyclicGroup(57);;Gdata:=PermutationRepForDiffsetCalculations(G);; gap> sigdat:=SignatureData(Gdata,[One(Gdata.G)],8,1,10^5); [ rec( subgroup := Group([ f2 ]), sigs := [ [ 1, 3, 4 ] ], cosets := [ [ 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54 ], [ 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56 ], [ 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 57 ] ] ) ]

`NormalSgsHavingAtMostNSigs(`

`sigdata``,`

`n``,`

`lengthlist``) F`

Let

`sigdata`be a list as returned by SignatureDataForNormalSubgroups, an integer`n`and a list of integers`lengthlist`.`NormalSgsHavingAtMostNSigs`

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``,`

`Gdata``,`

`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`Gdata`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`.The list of permutation groups

`autlist`is used for equivalence testing. Each equivalence test is performed calculating equivalence with respect to the first group, one element per equivalence class is retained and the equivalence test is repeated using the second group from`autlist`... Using an ascending list of automorphism groups can speed up the process of equivalence testing.gap> G:=CyclicGroup(57);;Gdata:=PermutationRepForDiffsetCalculations(G);; gap> sigdat:=SignatureData(Gdata,[One(Gdata.G)],8,1,10^5);; gap> N:=First(NormalSubgroups(G),n->Size(n)=19); gap> auts:=SuitableAutomorphismsForReduction(Gdata,N); [ <permutation group of size 18 with 3 generators> ] gap> g:=One(G);;while g in N do > g:=Random(G); > od; gap> coset:=GroupList2PermList(Set(RightCoset(N,g)),Gdata); [ 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56 ] gap> Size(StartsetsInCoset([],coset,[],4,auts,sigdat,Gdata,1)); #I Size 19 #I 1/ 0 @ 0:00:00.003 #I Size 26 #I 1/ 0 @ 0:00:00.001 #I -->10 @ 0:00:00.004 #I Size 88 #I 1/ 0 @ 0:00:00.003 #I -->45 @ 0:00:00.018 #I Size 125 #I 1/ 0 @ 0:00:00.006 #I -->64 @ 0:00:00.031 64 gap> Size(StartsetsInCoset([],coset,[],4,[Group(())],sigdat,Gdata,1)); #I Size 19 #I 1/ 0 @ 0:00:00.000 #I Size 136 #I 1/ 0 @ 0:00:00.004 #I -->136 @ 0:00:00.024 #I Size 648 #I 1/ 0 @ 0:00:00.021 #I -->648 @ 0:00:00.310 #I Size 1140 #I 1/ 0 @ 0:00:00.036 #I -->1140 @ 0:00:00.980 1140

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

February 2012