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# 2.3 Change of coset vs. brute force

Now we have startsets of length 2 in U and there are two possibilities:

(1) Find 3 more elements from another coset like this:

    gap> cosets:=RightCosets(G,U);
    [ RightCoset(Group( [ f1, f2, f3 ] ),<identity> of ...),
      RightCoset(Group( [ f1, f2, f3 ] ),f4),
      RightCoset(Group( [ f1, f2, f3 ] ),f4^2) ]
    gap> startsets:=StartsetsInCoset(startsets,cosets[2],N,5,auts,sigdat,Gdata,lambda);
    #I  Size 27
    #I  1/ 0 @ 0:00:00.632
    #I  Size 11
    #I  1/ 0 @ 0:00:00.260
    #I  Size 12
    #I  2/ 0 @ 0:00:00.340
    [ [ 4, 22, 5, 48, 59 ], [ 4, 22, 5, 59, 61 ] ]

And 3 more from the last one (of course, we could also change to force, but it seems to work this way...).

    gap> startsets:=StartsetsInCoset(startsets,cosets[3],N,8,auts,sigdat,Gdata,lambda);
    #I  Size 9
    #I  1/ 0 @ 0:00:00.300
    #I  Size 1
    #I  1/ 1 @ 0:00:00.024
    #I  Size 1
    #I  1/ 1 @ 0:00:00.028
    [ [ 4, 22, 5, 48, 59, 29, 72, 78 ] ]

So we found one difference set of order 9 in the elementary abelian group of order 81. To get the difference set containing 1 explicitly and as a subset of G, say

    gap> PermList2GroupList(Concatenation(startsets[1],[1]),Gdata);
    [ f3, f1*f3^2, f4, f2*f3^2*f4, f1*f2^2*f3*f4, f2*f4^2, f1^2*f3^2*f4^2,
    f1^2*f2^2*f3*f4^2, <identity> of ... ]

(2) Do a brute force search. Here we have to convert the forbidden group N into a list of integers Np. And we have to raise the length of the startsets by one before we can start. This is due to the ordering we chose (which is not necessarily compatible with the cosets modulo U).

    gap> Np:=GroupList2PermList(Set(N),Gdata);
    [ 1, 2, 3, 6, 7, 10, 16, 19, 32 ]
    gap> startsets:=ExtendedStartsetsNoSort(startsets,[1..groupOrder],Np,8,Gdata,lambda);;
    gap> Size(startsets);
    54
    gap> foundsets:=[];;
    gap> for set in startsets
    >  do
    >   Append(foundsets,AllDiffsets(set,[1..groupOrder],k-1,Np,Gdata,lambda));
    > od;
    gap> Size(foundsets);
    162

Now foundsets contains 162 relative (9,9,9,1)-difference sets (represented by lists of length 8).

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