Here is a similar example to that in chapter RDS:A quick start. But now we do a little more handwork to see how things work. Now we will calculate the relative difference sets of ``Dembowski-Piper type d'' and order 16. These difference sets represent projective planes which admit a quasiregular collineation group such that the fixed structure is an anti-flag. See DembowskiPiper, Dembowski or RoederDiss for details.
To have a little more output, you may want to increase InfoRDS
:
gap> SetInfoLevel(InfoRDS,3);
First, define some parameters and calculate the data needed:
gap> k:=16;;lambda:=1;;groupOrder:=255;; #Diffset parameters gap> forbiddenGroupOrder:=15;; gap> maxtest:=10^6;; #Bound for sig testing gap> G:=CyclicGroup(groupOrder); <pc group of size 255 with 3 generators> gap> Gdata:=PermutationRepForDiffsetCalculations(G);; gap> MakeImmutable(Gdata);;
Now the forbidden group is calculated in a very ineffective way. Then we calculate admissible signatures. As there are only few normal subgroups in G, we calculate them all. For other groups, one should choose more wisely.
gap> N:=First(NormalSubgroups(Gdata.G),i->Size(i)=forbiddenGroupOrder); Group([ f1*f3^9, f2*f3^10 ]) gap> globalSigData:=[];; gap> normals:=Filtered(NormalSubgroups(Gdata.G),n->Size(n) in [2..groupOrder-1]);; gap> sigdat:=SignatureDataForNormalSubgroups(normals,globalSigData, > N,Gdata,[k,lambda,maxtest,true]);;
The last step gives better results, if a larger maxtest is chosen.
But it also takes more time. To find a suitable maxtest, the output
of SignatureDataForNormalSubgroups
can be used, if
InfoLevel(InfoRDS)
is at least 2. Note that for recalculating
signatures, you will have to reset globalSigData to []
. Try experimenting
with maxtest to see the effect of signatures for the generation of
startsets.
Now examine the signatures found. Look if there is a normal subgroup
which has only one admissible signature (of course, you can also use
NormalSgsHavingAtMostNSigs
here):
gap> Set(Filtered(sigdat,s->Size(s.sigs)=1 and Size(s.sigs[1])<=5),i->i.sigs); [ [ [ 0, 4, 4, 4, 4 ] ], [ [ 4, 4, 8 ] ] ]
Great! we'll take the subgroup of index 3. The cosets modulo this group will be used to generate startsets and we assume that the trivial coset contains 8 elements of the difference set. So we generate startsets in U and make a first reduction. For this, we need U and N as lists of integers (recall that difference sets are asumed to be lists of integers). We will call these lists Up and Np. Furthermore, we will have to choose a suitable group of automorphisms for reduction. As G is cyclic, we may take Gdata·Aac here. A good choice is the stabilizer of all cosets modulo U. Yet sometimes larger groups may be possible. For example if we want to generate start sets in U and then do a brute force search. In this case, we may take the stabilizer of U for reduction.
gap> U:=First(sigdat,s->s.sigs=[ [ 4, 4, 8 ] ]).subgroup; Group([ f2, f3 ]) gap> cosets:=RightCosets(G,U); gap> U1:=cosets[2];;U2:=cosets[3];; gap> Up:=GroupList2PermList(Set(U),Gdata);; gap> Np:=GroupList2PermList(Set(N),Gdata); [ 1, 12, 25, 43, 78, 97, 115, 116, 134, 151, 169, 188, 207, 238, 249 ] gap> comps:=Difference(Up,Np);; gap> ssets:=List(comps,i->[i]);; gap> ssets:=ReducedStartsets(ssets,[Gdata.Aac],sigdat,Gdata.diffTable); #I Size 80 #I 2/ 0 @ 0:00:00.061 [ [ 3 ], [ 4 ] ]
Observe that 1 is assumed to be element of every difference set and
is not recorded explicitly. So the partial difference sets represented
by ssets at this point are tt[ [ 1, 3 ], [ 1, 4 ] ]. Now the
startsets are extended to size 7 using elements of Up. The runtime
varies depending on the output of SignatureDataForNormalSubgroups
and hence on maxtest.
gap> repeat > ssets:=ExtendedStartsets(ssets,comps,Np,7,Gdata); > ssets:=ReducedStartsets(ssets,[Gdata.Aac],sigdat,Gdata.diffTable);; > until ssets=[] or Size(ssets[1])=7; #I Size 133 #I 3/ 0 @ 0:00:00.133 #I Size 847 #I 4/ 0 @ 0:00:00.949 #I Size 6309 #I 4/ 0 @ 0:00:07.692 #I Size 21527 #I 5/ 0 @ 0:00:28.337 #I Size 15884 #I 4/ 0 @ 0:00:21.837 #I Size 1216 #I 4/ 0 @ 0:00:01.758 gap> Size(ssets); 192At a higher level of
InfoRDS
, the number of start sets which are
discarded because of wrong signatures are also shown.
Now the cosets are changed. Here we use the NoSort
version of
RaiseStartSetLevel
. This leads to a lot of start sets in the first
step. If the number of start sets in U is very large, this could be
too much for a reduction. Then the only option is using the brute
force method. But also for the brute force search,
RaiseStartSetLevelNoSort
must be called first (remember that we chos
an enumeration of G and assume the last element from each startset
to be the largeset ``interesting'' one for further completions).
gap> comps:=Difference(GroupList2PermList(Set(U1),Gdata),Np);; gap> ssets:=ExtendedStartsetsNoSort(ssets,comps,Np,11,Gdata);; gap> ssets:=ReducedStartsets(ssets,[Gdata.Aac],sigdat,Gdata.diffTable);; #I Size 8640 #I 9/ 0 @ 0:00:14.159 gap> Size(ssets); 6899And as above, we continue:
repeat ssets:=ExtendedStartsets(ssets,comps,Np,11,Gdata); ssets:=ReducedStartsets(ssets,[Gdata.Aac],sigdat,Gdata.diffTable);; until ssets=[] or Size(ssets[1])=11; comps:=Difference(GroupList2PermList(Set(U2),Gdata),Np); RaiseStartSetLevelNoSort(ssets,comps,Np,15,Gdata); repeat ssets:=ExtendedStartsets(ssets,comps,Np,15,Gdata); ssets:=ReducedStartsets(ssets,[Gdata.Aac],sigdat,Gdata.diffTable);; until ssets=[] or Size(ssets[1])=15;
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