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