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[Nam-Ky Nguyen]

> The calculation below (using the DESIGN Package for GAP) shows that
> there is no resolvable RGD with (v,b,r,k)=(8,18,9,4). 
Dear Leonard,

Thank your for your quick reply. Please give me a reference showing how
to calculate whether a resolvalbe RGD is available for a given
parameter  v,b,r and k. I study the output of your software below and I
cannot figure out how the calculation is done.

Regards,
Nam Nguyen

--------------------------------------------------------------------------
> gap> LoadPackage("design");
> 
> Loading  GRAPE 4.2  (GRaph Algorithms using PErmutation groups),
> by L.H.Soicher at qmul.ac.uk.
> 
>
-----------------------------------------------------------------------------
> Loading  DESIGN 1.1 (The Design Package for GAP)
> by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~leonard/).
>
-----------------------------------------------------------------------------
> true
> gap> v:=8;
> 8
> gap> k:=4;
> 4
> gap> gamma:=ComplementGraph(UnderlyingGraph(
> >          EdgeOrbitsGraph(Group(()),[[1,5],[2,6],[3,7],[4,8]],v) ));
> rec( isGraph := true, order := 8, group := Group(()), 
>   schreierVector := [ -1, -2, -3, -4, -5, -6, -7, -8 ], 
>   adjacencies := [ [ 2, 3, 4, 6, 7, 8 ], [ 1, 3, 4, 5, 7, 8 ], 
>       [ 1, 2, 4, 5, 6, 8 ], [ 1, 2, 3, 5, 6, 7 ], [ 2, 3, 4, 6, 7, 8
> ], 
>       [ 1, 3, 4, 5, 7, 8 ], [ 1, 2, 4, 5, 6, 8 ], [ 1, 2, 3, 5, 6, 7
> ] ], 
>   representatives := [ 1, 2, 3, 4, 5, 6, 7, 8 ], isSimple := true )
> gap> GlobalParameters(gamma);
> [ [ 0, 0, 6 ], [ 1, 4, 1 ], [ 6, 0, 0 ] ]
> gap> D:=BlockDesigns(rec(
> >     v:=v, blockSizes:=[k],
> >     tSubsetStructure:=rec(t:=2,
> >        partition:=[UndirectedEdges(ComplementGraph(gamma)),
> >                    UndirectedEdges(gamma)],
> >        lambdas:=[3,4])));;   
> gap> Length(D);
> 137
> gap> Runtime(); # in milliseconds
> 146875
> gap> Collected(List(D,AllTDesignLambdas));
> [ [ [ 18, 9 ], 137 ] ]
> gap> Collected(List(D,design->Size(AutomorphismGroup(design))));
> [ [ 1, 90 ], [ 2, 18 ], [ 3, 14 ], [ 4, 9 ], [ 6, 2 ], [ 8, 1 ], [
> 12, 1 ], 
>   [ 16, 1 ], [ 24, 1 ] ]
> gap> # Now determine whether any design in D is resolvable.
> gap> P:=List(D, design->PartitionsIntoBlockDesigns(rec(
> >      v:=v, blockSizes:=[k], blockDesign:=design,
> >      tSubsetStructure:=rec(t:=1,lambdas:=[1]))));;
> gap> Collected(P);
> [ [ [  ], 137 ] ]
> gap> Runtime();  # in milliseconds
> 149266
> gap> quit;
> 
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Nam-Ky Nguyen, Ph.D.
Design Computing
email: nkn at designcomputing.net
URL:   http://designcomputing.net/nkn/