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%
%A  construct.tex            DESIGN documentation              Leonard Soicher
%
%
%
\def\DESIGN{\sf DESIGN}
\def\GRAPE{\sf GRAPE}
\def\nauty{\it nauty}
\def\Aut{{\rm Aut}\,}
\def\lcm{{\rm lcm}\,}
\def\x{\times}
\Chapter{Constructing block designs}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{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 $\{1,\ldots,<v>\}$.

The first version of this function returns the block design with point-set
$\{1,\ldots,<v>\}$ 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 $\{1,\ldots,<v>\}$, and the function returns the
block design with point-set $\{1,\ldots,<v>\}$ and block multiset <C>,
where <C> is the sorted list of the concatenation of the <G>-orbits of
the elements of <B>.

\beginexample
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 ]
\endexample



\>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.

\beginexample
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 ]
\endexample



\>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.

\beginexample
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 ]
\endexample



\>WittDesign( <n> )

Suppose $<n>\in \{9,10,11,12,21,22,23,24\}$. 

If $<n>=24$ then this function returns the *large Witt design* $W_{24}$,
the unique (up to isomorphism) 5-(24,8,1) design. If $<n>=24-i$, where
$<i>\in \{1,2,3\}$, then the $i$-fold point-derived design of $W_{24}$
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* $W_{12}$,
the unique (up to isomorphism) 5-(12,6,1) design. If $<n>=12-i$, where
$<i>\in \{1,2,3\}$, then the $i$-fold point-derived design of $W_{12}$
is returned; this is the unique (up to isomorphism) $(5-i)$-$(12-i,6-i,1)$
design.

\beginexample
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)
\endexample



\>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>.

\beginexample
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 ] ]
\endexample



\>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).

\beginexample
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 ]
\endexample



\>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>.


\beginexample
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
\endexample



\>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>.

\beginexample
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 ] ] )
\endexample



\>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.

\beginexample
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> AddedPointBlockDesign(D,[[2,3,5],[2,3,5],[4,5,7]],"infinity");
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" ] )
\endexample



\>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.

\beginexample
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> AddedBlocksBlockDesign(D,D.blocks);
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 ] ] )
\endexample



\>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*
\index{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>.

\beginexample
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 ]
\endexample 



\>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*
\index{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>.

\beginexample
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 ]
\endexample 



\>TDesignFromTBD( <D>, <t>, <k> )

For $t$ a non-negative integer, $K$ a set of positive integers, and
$v,\lambda$ positive integers with $t\le v$, a $t$-*wise balanced design*,
or a $t$-$(v,K,\lambda)$ *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 $\lambda$ blocks.

Now let <t> and <k> be positive integers, <D> be a <t>-$(v,K,\lambda)$
design (for some set $K$), and $<t>\le <k>\le k_1$, where exactly $s$
distinct block-sizes $k_1\< \cdots \< k_s$ occur in <D>. Then this
function returns the <t>-design $D^{*}=D^{*}(t,k)$ described and studied
in \cite{McSo}.

The point set of $D^{*}$ is that of <D>, and the block multiset of $D^{*}$
consists of, for each $i=1,\ldots,s$ and each block $B$ of <D> of size
$k_i$ (including repeats), exactly $n/{k_i-t \choose k-t}$ copies of
every $k$-subset of $B$, where $n:=\lcm({k_i-t \choose k-t}:1\le i\le s)$.

It is shown in \cite{McSo} that $D^{*}$ is a $t$-$(v,k,n\lambda)$ design,
that $\Aut(D)\subseteq\Aut(D^{*})$, and that if $\lambda=1$ and $t\<k$,
then $\Aut(D)=\Aut(D^{*})$.

\beginexample
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 ]
\endexample