[rds] 2 AllDiffsets and OneDiffset

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# 2 AllDiffsets and OneDiffset

This chapter contains a number of examples as a very quick introduction to a few brute-force methods which can be used to find all (or just one) relative difference sets in a small group. Full documentation of these functions including all parameters can be found in section RDS:Brute force methods.

Do not expect too much from these methods alone! If you want to find examples of relative difference sets in larger groups, you should familiarize with the notion of coset signatures by also reading the next chapter.

The functions AllDiffsets and OneDiffset present the easiest way to calculate relative difference sets.

For a quick start, try this:

    gap> LoadPackage("rds");;
    gap> G:=CyclicGroup(7);
    <pc group of size 7 with 1 generators>
    gap> AllDiffsets(G);
    [ [ f1, f1^3 ], [ f1, f1^5 ], [ f1^2, f1^3 ], [ f1^2, f1^6 ], [ f1^4, f1^5 ],
      [ f1^4, f1^6 ] ]
    gap> OneDiffset(G);
    [ f1, f1^3 ]

The first is the set of all ordinary difference sets of order 2 in the cyclic group of order 7. Ok, they look too small (recall that the order of a difference set is the number k of elements it contains minus the multiplicity λ). Here is the reason:

Without loss of generality, every difference set contains the identity element of the group it lives in. RDS knows this and assumes it implicitly. So difference sets of length n are represented by lists of length n−1.

We can calculate all ordinary difference sets in G which contain the last element using AllDiffsetsNoSort. Observe, that AllDiffsets calculates partial difference sets by adding elements to the given list which are lexicographically larger than the last one of this list:

    gap> AllDiffsetsNoSort([Set(G)[7]],G);
    [ [ f1^6, f1^2 ], [ f1^6, f1^4 ] ]
    gap> AllDiffsets([Set(G)[7]],G);
    [  ]

You can also generate relative difference sets. Here we must give a partial difference set to start with (the empty list is ok) and a forbidden set. Notice that a forbidden subgroup cannot be input as a group. It has to be converted to a set.

    gap> G:=ElementaryAbelianGroup(81);
    <pc group of size 81 with 4 generators>
    gap> N:=Subgroup(G,GeneratorsOfGroup(G){[1,2]});
    Group([ f1, f2 ])
    gap> OneDiffset([],Set(N),G);
    [ f3, f4, f1*f3^2, f2*f3*f4, f1^2*f4^2, f2*f3^2*f4^2, f1*f2^2*f3^2*f4,
      f1^2*f2^2*f3*f4^2 ]

If the parameter λ is not given, it is set to 1. Of course, we can also find difference sets with λ > 1. Here is a (12,2,12,6) difference set in SL(2,3):

    gap> G:=SmallGroup(24,3);
    <pc group of size 24 with 4 generators>
    gap> N:=First(NormalSubgroups(G),i->Size(i)=2);
    Group([ f4 ])
    gap> OneDiffset([],Set(N),G,6);
    [ f1, f2, f3, f1^2, f1*f2, f1*f3, f2*f3, f1*f2*f3, f1^2*f2*f4, f1^2*f3*f4,
      f1^2*f2*f3*f4 ]

To test if a set is a relative difference set, IsDiffset can be used:

    gap> a:=(1,2,3,4,5,6,7);
    (1,2,3,4,5,6,7)
    gap> IsDiffset([a,a^3],Group(a));  #an ordinary difference set
    true
    gap> IsDiffset([a,a^2,a^4],Group(a));  #no ordinary difference set
    false
    gap> IsDiffset([a,a^2,a^4],Group(a),2);   #diffset with <lambda>=2
    true

In some cases, AllDiffsets and OneDiffset will refuse to work. A solution for this is to calculate `IsomorphismPermGroup` for your group and then work with the image under this isomorphism.

See RDS:Brute force methods for details.

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