Although the functions described in this section were initially meant to
investigate designs generated from nearrings, they can also be applied to
other incidence structures. In principal a design is represented as a set
of points and a set of blocks, a subset of the powerset of the points, with
containment as incidence relation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Constructing a design}
\>DesignFromPointsAndBlocks( <points>, <blocks> )
`DesignFromPointsAndBlocks' returns the design with the set of points
<points> and the set of blocks <blocks>, a subset of the powerset of
<points>.
\beginexample
gap> points := [1..7];;
gap> blocks := [[1,2,3],[1,4,5],[1,6,7],[2,4,7],[2,5,6],[3,5,7],
> [3,4,6]];;
gap> D := DesignFromPointsAndBlocks( points, blocks );
<an incidence structure with 7 points and 7 blocks> \endexample
\>DesignFromIncidenceMat( M );
`DesignFromIncidenceMat' returns the design with incidence matrix ,
The rows of <M> are labelled by the set of points $1$ to $v$, the
columns represent the blocks.
If the $( i, j )$ entry of the matrix <M> is $1$, then the point $i$ is
incident with the $j$-th block, i.e. the $j$-th block consists of those
points $i$ for which the entry $( i, j )$ of <M> is $1$. All other entries
have to be $0$.
\beginexample
gap> M := [[1,0,1,1],
> [1,1,0,0],
> [1,1,1,0]];;
gap> DesignFromIncidenceMat( M );
<an incidence structure with 3 points and 4 blocks> \endexample
\>DesignFromPlanarNearRing( <N>, <type> )
`DesignFromPlanarNearRing' returns a design obtained from the planar
nearring <N> following the constructions of James R. Clay \cite{Clay:Nearrings}.
If <type> = `"*"', `DesignFromPlanarNearRing' returns the design
$(<N>,B^{*},\in)$ in the notation of J. R. Clay with the elements of <N> as
set of points and $\{N^{*}\cdot a+b | a,b\in N, a \not= 0 \}$ as set of blocks.
Here $N^{*}$ is the set of elements $x\in N$ satisfying $x\cdot N = N$.
If <type> = `" "' (blank), `DesignFromPlanarNearRing' returns the design
$(<N>,B,\in)$ in the notation of J. R. Clay with the elements of <N> as
set of points and $\{N\cdot a+b | a,b\in N, a \not= 0 \}$ as set of
blocks.
`DesignFromFerreroPair' returns a design obtained from the group , and
a group of fixed-point-free automorphisms <phi> acting on <G> following the
constructions of James R. Clay \cite{Clay:Nearrings}.
If <type> = `"*"', `DesignFromFerreroPair' returns the design
$(<G>,B^{*},\in)$ in the notation of J. R. Clay with the elements of <G> as
set of points and the nonzero orbits of <G> under <phi> and their
translates by group-elements as set of blocks.
If <type> = `" "' (blank), `DesignFromFerreroPair' returns the design
$(<G>,B,\in)$ in the notation of J. R. Clay with the elements of <G> as
set of points and the nonzero orbits of <G> under <phi> joined with the zero
of <G> and their translates by group-elements as set of blocks.
\beginexample
gap> aux := FpfAutomorphismGroupsCyclic( [3,3], 4 );
[ [ [ f1, f2 ] -> [ f1*f2, f1*f2^2 ] ],
<pc group of size 9 with 2 generators> ]
gap> f := aux[1][1];
[ f1, f2 ] -> [ f1*f2, f1*f2^2 ]
gap> phi := Group( f );
<group with 1 generator>
gap> G := aux[2];
<pc group of size 9 with 2 generators>
gap> D3 := DesignFromFerreroPair( G, phi, "*" );
<a 2 - ( 9, 4, 3 ) nearring generated design>
gap> # D3 is actually isomorphic to D1 \endexample
\>DesignFromWdNearRing( <N> )
`DesignFromWdNearring' returns a design obtained from the weakly divisible
nearring <N> with cyclic additive group of prime power order. Following the
constructions of A. Benini, F. Morini and S. Pellegrini, we take the elements
of <N> as set of points and $\{N\cdot a+b | a\in C, b\in <N>\}$ as set of
blocks. Here $C$ is the set of elements $a\in N$ such that $a\cdot N = N$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Properties of a design}
\>PointsOfDesign( <D> )
`PointsOfDesign' returns the actual list of points of the design , not
their positions, no matter how the points of the design <D> may be
represented. To get the representation of those points, whose positions in
the list of all points are given by the list <pointnrs>, one can use
`PointsOfDesign( <D> )\{<pointnrs>\}'.
`BlocksOfDesign' returns the actual list of blocks of the design , not
their positions. Blocks are represented as lists of points. A point is
incident with a block if the point is an element of the
block. To get the representation of those blocks, whose positions in
the list of all blocks are given by the list <blocknrs>, one can use
`BlocksOfDesign( <D> )\{<blocknrs>\}'.
\beginexample
gap> Length( BlocksOfDesign( D1 ) );
18
gap> BlocksOfDesign( D1 ){[3]};
[ [ ((4,6,5)), (()), ((1,2,3)(4,5,6)), ((1,3,2)(4,5,6)) ] ]
gap> # returns the block in position 3 as a list of points \endexample
\>DesignParameter( <D> )
`DesignParameter' returns the set of paramaters $t, v, b, r, k, \lambda$
of the design <D>. Here $v$ is the size of the set of points
`PointsOfDesign', $b$ is the size of the set of blocks `PointsOfDesign',
every point is incident with precisely $r$ blocks, every block is incident
with precisely $k$ points, every $t$ distinct points are together incident
with precisely $\lambda$ blocks.
\beginexample
gap> DesignParameter( D1 );
[ 2, 9, 18, 8, 4, 3 ]
gap> # t = 2, v = 9, b = 18, r = 8, k = 4, lambda = 3 \endexample
\>IncidenceMat( <D> )
`IncidenceMat' returns the incidence matrix of the design , where the
rows are labelled by the positions of the points in `PointsOfDesign', the
columns are labelled by the positions of the blocks in `BlocksOfDesign'.
If the point in position $i$ is incident with the block in position $j$, then
the $( i, j )$ entry of the matrix `IncidenceMat' is $1$, else it is $0$.
`PrintIncidenceMat' prints only the entries of the incidence matrix
`IncidenceMat' of the design without commas. If the point in position $i$ is
incident with the block in position $j$, then there is $1$ in the $i$-th
row, $j$-th column, else there is '.', a dot.
In the first form `BlockIntersectionNumbers' returns the list of
cardinalities of the intersection of each block with all other blocks of the
design <D>.
In the second form `BlockIntersectionNumbers' returns the list of
cardinalities of the intersection of the block in position <blocknr> with
all other blocks of the design <D>.
\beginexample
gap> BlockIntersectionNumbers( D1, 2 );
[ 0, 4, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1 ]
gap> # the second has empty intersection with the first block
gap> # and intersects all others in at most 2 points \endexample
\>IsCircularDesign( <D> )
`IsCircularDesign' returns `true' if the design <D> is circular and `false'
otherwise. The design <D> has to be the result of `DesignFromPlanarNearRing'
or `DesignFromFerreroPair', since `IsCircularDesign' assumes the particular
structure of such a nearring-generated design.
A design <D> is *circular* if every two distinct blocks intersect in at most
two points. % CLAY'S DEFINITION IS TOO STRICT: % every three distinct points of <D> are incident with at most one block and % every two distinct points of <D> are incident with at least two distinct blocks.
`IsCircularDesign' calls the function `BlockIntersectionNumbers'.
`IsPointIncidentBlock' returns `true' if the point whose position in the list
`PointsOfDesign( <D> )' is given by is incident with the block
whose position in the list `BlocksOfDesign( <D> )' is given by ,
that is, the point is contained in the block as an element in a set.
\beginexample
gap> IsPointIncidentBlock( D1, 3, 1 );
true
gap> # point 3 is incident with block 1
gap> IsPointIncidentBlock( D1, 3, 2 );
false \endexample
\>PointsIncidentBlocks( <D>, <blocknrs> )
`PointsIncidentBlocks' returns a list of positions of those points of the
design <D> which are incident with the blocks, whose positions are given in
the list <blocknrs>.
\beginexample
gap> PointsIncidentBlocks( D1, [1, 4] );
[ 4, 7 ]
gap> # block 1 and block 4 are together incident with
gap> # points 4 and 7 \endexample
\>BlocksIncidentPoints( <D>, <pointnrs> )
`BlocksIncidentPoints' returns a list of positions of the blocks of the
design <D> which are incident with those points, whose positions are given
in the list <pointnrs>.
\beginexample
gap> BlocksIncidentPoints( D1, [2, 7] );
[ 1, 12, 15 ]
gap> # point 2 and point 7 are together incident with
gap> # blocks 1, 12, 15
gap> BlocksOfDesign( D1 ){last};
[ [ ((4,5,6)), ((4,6,5)), ((1,2,3)), ((1,3,2)) ],
[ ((1,3,2)), ((1,3,2)(4,5,6)), (()), ((4,5,6)) ],
[ ((1,3,2)(4,6,5)), ((1,3,2)), ((4,5,6)), ((1,2,3)(4,5,6)) ] ]
gap> # the actual point sets of blocks 1, 12, and 15
gap> BlocksIncidentPoints( D1, [2, 3, 7] );
[ 1 ]
gap> # points 2, 3, 7 are together incident with block 1
gap> PointsIncidentBlocks( D1, [1] );
[ 2, 3, 4, 7 ]
gap> # block 1 is incident with points 2, 3, 4, 7 \endexample
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