IsBlockDesign( obj )
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
IsBinaryBlockDesign( D )
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
IsSimpleBlockDesign( D )
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
IsConnectedBlockDesign( D )
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
BlockDesignPoints( D )
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 ]
NrBlockDesignPoints( D )
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
BlockDesignBlocks( D )
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 ] ]
NrBlockDesignBlocks( D )
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
BlockSizes( D )
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 ]
BlockNumbers( D )
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 ]
ReplicationNumber( D )
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
PairwiseBalancedLambda( D )
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
TSubsetLambdasVector( D, t )
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 ]
AllTDesignLambdas( D )
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 ]
AffineResolvableMu( D )
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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