4 Determining basic properties of block designs

This boolean function returns true if and only if obj, which can be an object of arbitrary type, is a block design.

gap> IsBlockDesign(5);
false
gap> IsBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) );
true

This boolean function returns true if and only if the block design D is binary, that is, if no block of D has a repeated element.

gap> IsBinaryBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) );
true
gap> IsBinaryBlockDesign( BlockDesign(2,[[1],[1,2],[1,2,2]]) );
false

This boolean function returns true if and only if the block design D is simple, that is, if no block of D is repeated.

gap> IsSimpleBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) );  
false
gap> IsSimpleBlockDesign( BlockDesign(2,[[1],[1,2],[1,2,2]]) );
true

This boolean function returns true if and only if the block design D is connected, that is, if its incidence graph is a connected graph.

gap> IsConnectedBlockDesign( BlockDesign(2,[[1],[2]]) ); 
false
gap> IsConnectedBlockDesign( BlockDesign(2,[[1,2]]) );  
true

This function returns the set of points of the block design D, that is [1..D.v]. The returned result is immutable.

gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]);
rec( isBlockDesign := true, v := 3, 
  blocks := [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] )
gap> BlockDesignPoints(D);                       
[ 1 .. 3 ]

This function returns the number of points of the block design D.

gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]);
rec( isBlockDesign := true, v := 3, 
  blocks := [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] )
gap> NrBlockDesignPoints(D);                     
3

This function returns the (sorted) list of blocks of the block design D. The returned result is immutable.

gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]);
rec( isBlockDesign := true, v := 3, 
  blocks := [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] )
gap> BlockDesignBlocks(D);                       
[ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ]

This function returns the number of blocks of the block design D.

gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]);
rec( isBlockDesign := true, v := 3, 
  blocks := [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] )
gap> NrBlockDesignBlocks(D);                     
4

This function returns the set of sizes (actually list-lengths) of the blocks of the block design D.

gap> BlockSizes( BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]) );  
[ 1, 3 ]

Let D be a block design. Then this function returns a list of the same length as BlockSizes(D), such that the i-th element of this returned list is the number of blocks of D of size BlockSizes(D)[i].

gap> D:=BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]); 
rec( isBlockDesign := true, v := 3, 
  blocks := [ [ 1 ], [ 1, 2, 2 ], [ 1, 2, 3 ], [ 2 ], [ 3 ] ] )
gap> BlockSizes(D);
[ 1, 3 ]
gap> BlockNumbers(D);
[ 3, 2 ]

If the block design D is equireplicate, then this function returns its replication number; otherwise fail is returned.

A block design D is equireplicate with replication number r if, for every point x of D, r is equal to the sum over the blocks of the multiplicity of x in a block. For a binary block design this is the same as saying that each point x is contained in exactly r blocks.

gap> ReplicationNumber(BlockDesign(4,[[1],[1,2],[2,3,3],[4,4]]));
2
gap> ReplicationNumber(BlockDesign(4,[[1],[1,2],[2,3],[4,4]]));  
fail

A binary block design D is pairwise balanced if D has at least two points and every pair of distinct points is contained in exactly lambda blocks, for some positive constant lambda.

Given a binary block design D, this function returns fail if D is not pairwise balanced, and otherwise the positive constant lambda such that every pair of distinct points of D is in exactly lambda blocks.

gap> D:=BlockDesigns(rec(v:=10, blockSizes:=[3,4],       
>          tSubsetStructure:=rec(t:=2,lambdas:=[1])))[1];
rec( isBlockDesign := true, v := 10, 
  blocks := [ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 10 ], 
      [ 2, 6, 8 ], [ 2, 7, 9 ], [ 3, 5, 9 ], [ 3, 6, 10 ], [ 3, 7, 8 ], 
      [ 4, 5, 8 ], [ 4, 6, 9 ], [ 4, 7, 10 ] ], 
  tSubsetStructure := rec( t := 2, lambdas := [ 1 ] ), isBinary := true, 
  isSimple := true, blockSizes := [ 3, 4 ], blockNumbers := [ 9, 3 ], 
  autGroup := Group([ (5,6,7)(8,9,10), (2,3)(5,7)(8,10), 
      (2,3,4)(5,7,6)(8,9,10), (2,3,4)(5,9,6,8,7,10), (2,6,9,3,7,10)(4,5,8) ]) 
 )
gap> PairwiseBalancedLambda(D);                          
1

Let D be a block design, t a non-negative integer, and v=D.v. Then this function returns an integer vector L whose positions correspond to the t-subsets of {1,...,v}. The i-th element of L is the sum over all blocks B of D of the number of times the i-th t-subset (in lexicographic order) is contained in B. (For example, if t=2 and B=[1,1,2,3,3,4], then B contains [1,2] twice, [1,3] four times, [1,4] twice, [2,3] twice, [2,4] once, and [3,4] twice.) In particular, if D is binary then L[i] is simply the number of blocks of D containing the i-th t-subset (in lexicographic order).

gap> D:=BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]);;
gap> TSubsetLambdasVector(D,0);
[ 5 ]
gap> TSubsetLambdasVector(D,1);
[ 3, 4, 2 ]
gap> TSubsetLambdasVector(D,2);
[ 3, 1, 1 ]
gap> TSubsetLambdasVector(D,3);
[ 1 ]

If the block design D is not a t-design for some tge0 then this function returns an empty list. Otherwise D is a binary block design with constant block size k, say, and this function returns a list L of length T+1, where T is the maximum tlek such that D is a t-design, and, for i=1,...,T+1, L[i] is equal to the (constant) number of blocks of D containing an (i-1)-subset of the point-set of D. The returned result is immutable.

gap> AllTDesignLambdas(PGPointFlatBlockDesign(3,2,1));                  
[ 35, 7, 1 ]

A block design is affine resolvable if the design is resolvable and any two blocks not in the same parallel class of a resolution meet in a constant number mu of points.

If the block design D is affine resolvable, then this function returns its value of mu; otherwise fail is returned.

The value 0 is returned if, and only if, D consists of a single parallel class.

gap> P:=PGPointFlatBlockDesign(2,3,1);; # projective plane of order 3 
gap> AffineResolvableMu(P);
fail
gap> A:=ResidualBlockDesign(P,P.blocks[1]);; # affine plane of order 3
gap> AffineResolvableMu(A);                                           
1

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