Here is a similar example to that in chapter RDS:A basic example. 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
ExtendedStartsets. This leads to a lot of start sets in the first
step. If the number of start sets in

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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rds manual

February 2022