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# 3 Constructing block designs

## 3.1 Functions to construct block designs

• `BlockDesign( `v`, `B` )`
• `BlockDesign( `v`, `B`, `G` )`

Let v be a positive integer and B a non-empty list of non-empty sorted lists of elements of {1,…,v }.

The first version of this function returns the block design with point-set {1,…,v } and block multiset C, where C is `SortedList(`B`)`.

For the second version of this function, we require G to be a group of permutations of {1,…,v }, and the function returns the block design with point-set {1,…,v } and block multiset C, where C is the sorted list of the concatenation of the G-orbits of the elements of B.

```gap> BlockDesign( 2, [[1,2],[1],[1,2]] );
rec( isBlockDesign := true, v := 2, blocks := [ [ 1 ], [ 1, 2 ], [ 1, 2 ] ] )
gap> D:=BlockDesign(7, [[1,2,4]], Group((1,2,3,4,5,6,7)));
rec( isBlockDesign := true, v := 7,
blocks := [ [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 5, 6 ], [ 2, 3, 5 ],
[ 2, 6, 7 ], [ 3, 4, 6 ], [ 4, 5, 7 ] ],
autSubgroup := Group([ (1,2,3,4,5,6,7) ]) )
gap> AllTDesignLambdas(D);
[ 7, 3, 1 ]
```

• `AGPointFlatBlockDesign( `n`, `q`, `d` )`

Let n be positive integer, q a prime-power, and d a non-negative integer less than or equal to n. Then this function returns the block design whose points are the points of the affine space AG(n ,q ), and whose blocks are the d -flats of AG(n ,q ), considering a d -flat as a set of points.

Note that the affine space AG(n ,q ) consists of all the cosets of all the subspaces of the vector space V(n ,q ), with the points being the cosets of the 0-dimensional subspace and the d -flats being the cosets of the d -dimensional subspaces. As is usual, we identify the points with the vectors in V(n ,q ), and these vectors are given as the point-names.

```gap> D:=AGPointFlatBlockDesign(2,4,1);
rec( isBlockDesign := true, v := 16,
blocks := [ [ 1, 2, 3, 4 ], [ 1, 5, 9, 13 ], [ 1, 6, 11, 16 ],
[ 1, 7, 12, 14 ], [ 1, 8, 10, 15 ], [ 2, 5, 12, 15 ], [ 2, 6, 10, 14 ],
[ 2, 7, 9, 16 ], [ 2, 8, 11, 13 ], [ 3, 5, 10, 16 ], [ 3, 6, 12, 13 ],
[ 3, 7, 11, 15 ], [ 3, 8, 9, 14 ], [ 4, 5, 11, 14 ], [ 4, 6, 9, 15 ],
[ 4, 7, 10, 13 ], [ 4, 8, 12, 16 ], [ 5, 6, 7, 8 ], [ 9, 10, 11, 12 ],
[ 13, 14, 15, 16 ] ],
autSubgroup := Group([ (5,9,13)(6,10,14)(7,11,15)(8,12,16),
(2,5,6)(3,9,11)(4,13,16)(7,14,12)(8,10,15),
(1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16),
(3,4)(7,8)(9,13)(10,14)(11,16)(12,15) ]),
pointNames := [ [ 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2^2) ],
[ 0*Z(2), Z(2^2)^2 ], [ Z(2)^0, 0*Z(2) ], [ Z(2)^0, Z(2)^0 ],
[ Z(2)^0, Z(2^2) ], [ Z(2)^0, Z(2^2)^2 ], [ Z(2^2), 0*Z(2) ],
[ Z(2^2), Z(2)^0 ], [ Z(2^2), Z(2^2) ], [ Z(2^2), Z(2^2)^2 ],
[ Z(2^2)^2, 0*Z(2) ], [ Z(2^2)^2, Z(2)^0 ], [ Z(2^2)^2, Z(2^2) ],
[ Z(2^2)^2, Z(2^2)^2 ] ] )
gap> AllTDesignLambdas(D);
[ 20, 5, 1 ]
```

• `PGPointFlatBlockDesign( `n`, `q`, `d` )`

Let n be a non-negative integer, q a prime-power, and d a non-negative integer less than or equal to n. Then this function returns the block design whose points are the (projective) points of the projective space PG(n ,q ), and whose blocks are the d -flats of PG(n ,q ), considering a d -flat as a set of projective points.

Note that the projective space PG(n ,q ) consists of all the subspaces of the vector space V(n +1,q ), with the projective points being the 1-dimensional subspaces and the d -flats being the (d +1)-dimensional subspaces.

```gap> D:=PGPointFlatBlockDesign(3,2,1);
rec( isBlockDesign := true, v := 15,
blocks := [ [ 1, 2, 3 ], [ 1, 4, 5 ], [ 1, 6, 7 ], [ 1, 8, 9 ],
[ 1, 10, 11 ], [ 1, 12, 13 ], [ 1, 14, 15 ], [ 2, 4, 6 ], [ 2, 5, 7 ],
[ 2, 8, 10 ], [ 2, 9, 11 ], [ 2, 12, 14 ], [ 2, 13, 15 ], [ 3, 4, 7 ],
[ 3, 5, 6 ], [ 3, 8, 11 ], [ 3, 9, 10 ], [ 3, 12, 15 ], [ 3, 13, 14 ],
[ 4, 8, 12 ], [ 4, 9, 13 ], [ 4, 10, 14 ], [ 4, 11, 15 ], [ 5, 8, 13 ],
[ 5, 9, 12 ], [ 5, 10, 15 ], [ 5, 11, 14 ], [ 6, 8, 14 ], [ 6, 9, 15 ],
[ 6, 10, 12 ], [ 6, 11, 13 ], [ 7, 8, 15 ], [ 7, 9, 14 ],
[ 7, 10, 13 ], [ 7, 11, 12 ] ],
autSubgroup := Group([ (8,12)(9,13)(10,14)(11,15),
(1,2,4,8)(3,6,12,9)(5,10)(7,14,13,11) ]),
pointNames := [ <vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)> ] )
gap> AllTDesignLambdas(D);
[ 35, 7, 1 ]
```

• `WittDesign( `n` )`

Suppose n ∈ {9,10,11,12,21,22,23,24}.

If n =24 then this function returns the large Witt design W24, the unique (up to isomorphism) 5-(24,8,1) design. If n =24−i, where i ∈ {1,2,3}, then the i-fold point-derived design of W24 is returned; this is the unique (up to isomorphism) (5−i)-(24−i,8−i,1) design.

If n =12 then this function returns the small Witt design W12, the unique (up to isomorphism) 5-(12,6,1) design. If n =12−i, where i ∈ {1,2,3}, then the i-fold point-derived design of W12 is returned; this is the unique (up to isomorphism) (5−i)-(12−i,6−i,1) design.

```gap> W24:=WittDesign(24);;
gap> AllTDesignLambdas(W24);
[ 759, 253, 77, 21, 5, 1 ]
gap> DisplayCompositionSeries(AutomorphismGroup(W24));
G (3 gens, size 244823040)
| M(24)
1 (0 gens, size 1)
gap> W10:=WittDesign(10);;
gap> AllTDesignLambdas(W10);
[ 30, 12, 4, 1 ]
gap> DisplayCompositionSeries(AutomorphismGroup(W10));
G (4 gens, size 1440)
| Z(2)
S (4 gens, size 720)
| Z(2)
S (3 gens, size 360)
| A(6) ~ A(1,9) = L(2,9) ~ B(1,9) = O(3,9) ~ C(1,9) = S(2,9) ~ 2A(1,9) = U(2,\
9)
1 (0 gens, size 1)
```

• `DualBlockDesign( `D` )`

Suppose D is a block design for which every point lies on at least one block. Then this function returns the dual of D, the block design in which the roles of points and blocks are interchanged, but incidence (including repeated incidence) stays the same. Note that, since the list of blocks of a block design is always sorted, the block list of the dual of the dual of D may not be equal to the block list of D.

```gap> D:=BlockDesign(4,[[1,3],[2,3,4],[3,4]]);;
gap> dualD:=DualBlockDesign(D);
rec( isBlockDesign := true, v := 3,
blocks := [ [ 1 ], [ 1, 2, 3 ], [ 2 ], [ 2, 3 ] ],
pointNames := [ [ 1, 3 ], [ 2, 3, 4 ], [ 3, 4 ] ] )
gap> DualBlockDesign(dualD).blocks;
[ [ 1, 2 ], [ 2, 3, 4 ], [ 2, 4 ] ]
```

• `ComplementBlocksBlockDesign( `D` )`

Suppose D is a binary incomplete-block design. Then this function returns the block design on the same point-set as D, whose blocks are the complements of those of D (complemented with respect to the point-set).

```gap> D:=PGPointFlatBlockDesign(2,2,1);
rec( isBlockDesign := true, v := 7,
pointNames := [ <vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)> ],
blocks := [ [ 1, 2, 3 ], [ 1, 4, 5 ], [ 1, 6, 7 ], [ 2, 4, 6 ],
[ 2, 5, 7 ], [ 3, 4, 7 ], [ 3, 5, 6 ] ] )
gap> AllTDesignLambdas(D);
[ 7, 3, 1 ]
gap> C:=ComplementBlocksBlockDesign(D);
rec( isBlockDesign := true, v := 7,
blocks := [ [ 1, 2, 4, 7 ], [ 1, 2, 5, 6 ], [ 1, 3, 4, 6 ], [ 1, 3, 5, 7 ],
[ 2, 3, 4, 5 ], [ 2, 3, 6, 7 ], [ 4, 5, 6, 7 ] ],
pointNames := [ <vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)>,
<vector space of dimension 1 over GF(2)> ] )
gap> AllTDesignLambdas(C);
[ 7, 4, 2 ]
```

• `DeletedPointsBlockDesign( `D`, `Y` )`

Suppose D is a block design and Y is a proper subset of the point-set of D.

Then this function returns the block design DP obtained from D by deleting the points in Y from the point-set, and from each block. It is an error if the resulting design contains an empty block. The points of DP are relabelled 1,2,·.·, preserving the order of the corresponding points of D; the point-names of DP (listed in `DP.pointNames`) are those of these corresponding points of D.

```gap> D:=BlockDesigns(rec(v:=11,blockSizes:=[5],
>       tSubsetStructure:=rec(t:=2,lambdas:=[2])))[1];
rec( isBlockDesign := true, v := 11,
blocks := [ [ 1, 2, 3, 4, 5 ], [ 1, 2, 9, 10, 11 ], [ 1, 3, 6, 7, 9 ],
[ 1, 4, 7, 8, 10 ], [ 1, 5, 6, 8, 11 ], [ 2, 3, 6, 8, 10 ],
[ 2, 4, 6, 7, 11 ], [ 2, 5, 7, 8, 9 ], [ 3, 4, 8, 9, 11 ],
[ 3, 5, 7, 10, 11 ], [ 4, 5, 6, 9, 10 ] ],
tSubsetStructure := rec( t := 2, lambdas := [ 2 ] ), isBinary := true,
isSimple := true, blockSizes := [ 5 ], blockNumbers := [ 11 ], r := 5,
autGroup := Group([ (2,4)(3,5)(7,11)(8,9), (1,3)(2,5)(7,9)(10,11),
(1,5,3)(6,11,7)(8,10,9), (1,10,5,2,11,3)(4,9,7)(6,8) ]) )
gap> AllTDesignLambdas(D);
[ 11, 5, 2 ]
gap> DP:=DeletedPointsBlockDesign(D,[5,8]);
rec( isBlockDesign := true, v := 9,
blocks := [ [ 1, 2, 3, 4 ], [ 1, 2, 7, 8, 9 ], [ 1, 3, 5, 6, 7 ],
[ 1, 4, 6, 8 ], [ 1, 5, 9 ], [ 2, 3, 5, 8 ], [ 2, 4, 5, 6, 9 ],
[ 2, 6, 7 ], [ 3, 4, 7, 9 ], [ 3, 6, 8, 9 ], [ 4, 5, 7, 8 ] ],
pointNames := [ 1, 2, 3, 4, 6, 7, 9, 10, 11 ] )
gap> PairwiseBalancedLambda(DP);
2
```

• `DeletedBlocksBlockDesign( `D`, `Y`)`

Suppose D is a block design, and Y is a proper sublist of the block-list of D (Y need not be sorted).

Then this function returns the block design obtained from D by deleting the blocks in Y (counting repeats) from the block-list of D.

```gap> D:=BlockDesign(7,[[1,2,4],[1,2,4]],Group((1,2,3,4,5,6,7)));
rec( isBlockDesign := true, v := 7,
blocks := [ [ 1, 2, 4 ], [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 3, 7 ],
[ 1, 5, 6 ], [ 1, 5, 6 ], [ 2, 3, 5 ], [ 2, 3, 5 ], [ 2, 6, 7 ],
[ 2, 6, 7 ], [ 3, 4, 6 ], [ 3, 4, 6 ], [ 4, 5, 7 ], [ 4, 5, 7 ] ],
autSubgroup := Group([ (1,2,3,4,5,6,7) ]) )
gap> DeletedBlocksBlockDesign(D,[[2,3,5],[2,3,5],[4,5,7]]);
rec( isBlockDesign := true, v := 7,
blocks := [ [ 1, 2, 4 ], [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 3, 7 ],
[ 1, 5, 6 ], [ 1, 5, 6 ], [ 2, 6, 7 ], [ 2, 6, 7 ], [ 3, 4, 6 ],
[ 3, 4, 6 ], [ 4, 5, 7 ] ] )
```

• `AddedPointBlockDesign( `D`, `Y` )`
• `AddedPointBlockDesign( `D`, `Y`, `pointname` )`

Suppose D is a block design, and Y is a sublist of the block-list of D (Y need not be sorted).

Then this function returns the block design obtained from D by adding the new point D`.v+1` to the point-set, and adding this new point (once) to each block of Y (where repeats count).

The optional parameter pointname specifies a point-name for the new point.

```gap> D:=BlockDesign(7,[[1,2,4],[1,2,4]],Group((1,2,3,4,5,6,7)));
rec( isBlockDesign := true, v := 7,
blocks := [ [ 1, 2, 4 ], [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 3, 7 ],
[ 1, 5, 6 ], [ 1, 5, 6 ], [ 2, 3, 5 ], [ 2, 3, 5 ], [ 2, 6, 7 ],
[ 2, 6, 7 ], [ 3, 4, 6 ], [ 3, 4, 6 ], [ 4, 5, 7 ], [ 4, 5, 7 ] ],
autSubgroup := Group([ (1,2,3,4,5,6,7) ]) )
rec( isBlockDesign := true, v := 8,
blocks := [ [ 1, 2, 4 ], [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 3, 7 ],
[ 1, 5, 6 ], [ 1, 5, 6 ], [ 2, 3, 5, 8 ], [ 2, 3, 5, 8 ], [ 2, 6, 7 ],
[ 2, 6, 7 ], [ 3, 4, 6 ], [ 3, 4, 6 ], [ 4, 5, 7 ], [ 4, 5, 7, 8 ] ],
pointNames := [ 1, 2, 3, 4, 5, 6, 7, "infinity" ] )
```

• `AddedBlocksBlockDesign( `D`, `Y` )`

Suppose Y is a list of multisets of points of the block design D. Then this function returns a new block design, whose point-set is that of D, and whose block list is that of D with the elements of Y (including repeats) added.

```gap> D:=BlockDesign(7,[[1,2,4]],Group((1,2,3,4,5,6,7)));
rec( isBlockDesign := true, v := 7,
blocks := [ [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 5, 6 ], [ 2, 3, 5 ],
[ 2, 6, 7 ], [ 3, 4, 6 ], [ 4, 5, 7 ] ],
autSubgroup := Group([ (1,2,3,4,5,6,7) ]) )
rec( isBlockDesign := true, v := 7,
blocks := [ [ 1, 2, 4 ], [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 3, 7 ],
[ 1, 5, 6 ], [ 1, 5, 6 ], [ 2, 3, 5 ], [ 2, 3, 5 ], [ 2, 6, 7 ],
[ 2, 6, 7 ], [ 3, 4, 6 ], [ 3, 4, 6 ], [ 4, 5, 7 ], [ 4, 5, 7 ] ] )
```

• `DerivedBlockDesign( `D`, `x` )`

Suppose D is a block design, and x is a point or block of D. Then this function returns the derived design DD of D, with respect to x.

If x is a point then DD is the block design whose blocks are those of D containing x, but with x deleted from these blocks, and the points of DD are those which occur in some block of DD.

If x is a block, then the points of DD are the points in x, and the blocks of DD are the blocks of D other than x containing at least one point of x, but with all points not in x deleted from these blocks. Note that any repeat of x, but not x itself, is a block of DD.

It is an error if the resulting block design DD has no blocks or an empty block.

The points of DD are relabelled 1,2,·.·, preserving the order of the corresponding points of D; the point-names of DD (listed in `DD.pointNames`) are those of these corresponding points of D.

```gap> D:=BlockDesigns(rec(v:=11,blockSizes:=[5],
>       tSubsetStructure:=rec(t:=2,lambdas:=[2])))[1];
rec( isBlockDesign := true, v := 11,
blocks := [ [ 1, 2, 3, 4, 5 ], [ 1, 2, 9, 10, 11 ], [ 1, 3, 6, 7, 9 ],
[ 1, 4, 7, 8, 10 ], [ 1, 5, 6, 8, 11 ], [ 2, 3, 6, 8, 10 ],
[ 2, 4, 6, 7, 11 ], [ 2, 5, 7, 8, 9 ], [ 3, 4, 8, 9, 11 ],
[ 3, 5, 7, 10, 11 ], [ 4, 5, 6, 9, 10 ] ],
tSubsetStructure := rec( t := 2, lambdas := [ 2 ] ), isBinary := true,
isSimple := true, blockSizes := [ 5 ], blockNumbers := [ 11 ], r := 5,
autGroup := Group([ (2,4)(3,5)(7,11)(8,9), (1,3)(2,5)(7,9)(10,11),
(1,5,3)(6,11,7)(8,10,9), (1,10,5,2,11,3)(4,9,7)(6,8) ]) )
gap> AllTDesignLambdas(D);
[ 11, 5, 2 ]
gap> DD:=DerivedBlockDesign(D,6);
rec( isBlockDesign := true, v := 10,
blocks := [ [ 1, 3, 6, 8 ], [ 1, 5, 7, 10 ], [ 2, 3, 7, 9 ],
[ 2, 4, 6, 10 ], [ 4, 5, 8, 9 ] ],
pointNames := [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 11 ] )
gap> AllTDesignLambdas(DD);
[ 5, 2 ]
gap> DD:=DerivedBlockDesign(D,D.blocks[6]);
rec( isBlockDesign := true, v := 5,
blocks := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 3 ], [ 2, 4 ],
[ 2, 5 ], [ 3, 4 ], [ 3, 5 ], [ 4, 5 ] ],
pointNames := [ 2, 3, 6, 8, 10 ] )
gap> AllTDesignLambdas(DD);
[ 10, 4, 1 ]
```

• `ResidualBlockDesign( `D`, `x` )`

Suppose D is a block design, and x is a point or block of D. Then this function returns the residual design RD of D, with respect to x.

If x is a point then RD is the block design whose blocks are those of D not containing x, and the points of RD are those which occur in some block of RD.

If x is a block, then the points of RD are those of D not in x, and the blocks of RD are the blocks of D (including repeats) containing at least one point not in x, but with all points in x deleted from these blocks.

It is an error if the resulting block design RD has no blocks.

The points of RD are relabelled 1,2,·.·, preserving the order of the corresponding points of D; the point-names of RD (listed in `RD.pointNames`) are those of these corresponding points of D.

```gap> D:=BlockDesigns(rec(v:=11,blockSizes:=[5],
>       tSubsetStructure:=rec(t:=2,lambdas:=[2])))[1];
rec( isBlockDesign := true, v := 11,
blocks := [ [ 1, 2, 3, 4, 5 ], [ 1, 2, 9, 10, 11 ], [ 1, 3, 6, 7, 9 ],
[ 1, 4, 7, 8, 10 ], [ 1, 5, 6, 8, 11 ], [ 2, 3, 6, 8, 10 ],
[ 2, 4, 6, 7, 11 ], [ 2, 5, 7, 8, 9 ], [ 3, 4, 8, 9, 11 ],
[ 3, 5, 7, 10, 11 ], [ 4, 5, 6, 9, 10 ] ],
tSubsetStructure := rec( t := 2, lambdas := [ 2 ] ), isBinary := true,
isSimple := true, blockSizes := [ 5 ], blockNumbers := [ 11 ], r := 5,
autGroup := Group([ (2,4)(3,5)(7,11)(8,9), (1,3)(2,5)(7,9)(10,11),
(1,5,3)(6,11,7)(8,10,9), (1,10,5,2,11,3)(4,9,7)(6,8) ]) )
gap> AllTDesignLambdas(D);
[ 11, 5, 2 ]
gap> RD:=ResidualBlockDesign(D,6);
rec( isBlockDesign := true, v := 10,
blocks := [ [ 1, 2, 3, 4, 5 ], [ 1, 2, 8, 9, 10 ], [ 1, 4, 6, 7, 9 ],
[ 2, 5, 6, 7, 8 ], [ 3, 4, 7, 8, 10 ], [ 3, 5, 6, 9, 10 ] ],
pointNames := [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 11 ] )
gap> AllTDesignLambdas(RD);
[ 6, 3 ]
gap> RD:=ResidualBlockDesign(D,D.blocks[6]);
rec( isBlockDesign := true, v := 6,
blocks := [ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 6 ], [ 1, 4, 5 ],
[ 1, 5, 6 ], [ 2, 3, 5 ], [ 2, 4, 6 ], [ 2, 5, 6 ], [ 3, 4, 5 ],
[ 3, 4, 6 ] ], pointNames := [ 1, 4, 5, 7, 9, 11 ] )
gap> AllTDesignLambdas(RD);
[ 10, 5, 2 ]
```

• `TDesignFromTBD( `D`, `t`, `k` )`

For t a non-negative integer, K a set of positive integers, and v,λ positive integers with tv, a t-wise balanced design, or a t-(v,K,λ) design, is a binary block design with exactly v points, such that each block has size in K and each t-subset of the points is contained in exactly λ blocks.

Now let t and k be positive integers, D be a t-(v,K,λ) design (for some set K), and tkk1, where exactly s distinct block-sizes k1 < … < ks occur in D. Then this function returns the t-design D*=D*(t,k) described and studied in McSo.

The point set of D* is that of D, and the block multiset of D* consists of, for each i=1,…,s and each block B of D of size ki (including repeats), exactly
 n/ ⎛⎝ ki−tk−t ⎞⎠
copies of every k-subset of B, where
 n:=\lcm( ⎛⎝ ki−tk−t ⎞⎠ :1 ≤ i ≤ s)
.

It is shown in McSo that D* is a t-(v,k,nλ) design, that \Aut(D) ⊆ \Aut(D*), and that if λ = 1 and t < k, then \Aut(D)=\Aut(D*).

```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
gap> Dstar:=TDesignFromTBD(D,2,3);
rec( isBlockDesign := true, v := 10,
blocks := [ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 4 ], [ 1, 5, 6 ],
[ 1, 5, 7 ], [ 1, 6, 7 ], [ 1, 8, 9 ], [ 1, 8, 10 ], [ 1, 9, 10 ],
[ 2, 3, 4 ], [ 2, 5, 10 ], [ 2, 5, 10 ], [ 2, 6, 8 ], [ 2, 6, 8 ],
[ 2, 7, 9 ], [ 2, 7, 9 ], [ 3, 5, 9 ], [ 3, 5, 9 ], [ 3, 6, 10 ],
[ 3, 6, 10 ], [ 3, 7, 8 ], [ 3, 7, 8 ], [ 4, 5, 8 ], [ 4, 5, 8 ],
[ 4, 6, 9 ], [ 4, 6, 9 ], [ 4, 7, 10 ], [ 4, 7, 10 ], [ 5, 6, 7 ],
[ 8, 9, 10 ] ],
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> AllTDesignLambdas(Dstar);
[ 30, 9, 2 ]
```

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design manual
March 2019