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# 4 Determining basic properties of block designs

### Sections

1. The functions for basic properties

## 4.1 The functions for basic properties

- `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 λ blocks, for some positive constant λ.

  Given a binary block design D, this function returns `fail` if D is not pairwise balanced, and otherwise the positive constant λ such that every pair of distinct points of D is in exactly λ 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 t ≥ 0 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 t ≤ k 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 μ of points.

  If the block design D is affine resolvable, then this function returns its value of μ; 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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