The GAP packages UnitalSZ and GRAPE are necessary.
gap> LoadPackage( "UnitalSZ", false ); gap> LoadPackage( "grape", false );
You can use the info class InfoParamod to get extra information.
gap> DeclareInfoClass( "InfoParamod" );
‣ AllRegularBlockColorings( bls, k, gr ) | ( function ) |
Returns: the list of block colorings of the set bls of blocks, with k colors and color classes of size |bls|/k. The colorings are returned as GAP transformation objects from [1..Size(bls)] to [1..k].
Let \(P\) be a set, \(B\) a set of subsets (called blocks), and \(k\) a positive integer. The map \(\chi:B\to \{1,\ldots,k\}\) is a block coloring if \(\chi(b)=\chi(b')\) implies \(b\cap b'=\emptyset\) for any \(b,b' \in B\). The block coloring in regular, if each color class has size \(|B|/k\). The last argument gr must be a block preserving permutation of the set \(P\) of points.
‣ ParamodificationOfUnital( u, b, chi ) | ( function ) |
Returns: the paramodification of the unital u with respect to the block b and regular block coloring chi with |b| colors.
The coloring chi must be given as a GAP transformation object from [1..q^2*(q^2-q+1)] to [1..q+1], where q is the order of u. This function has a slightly faster non-checking version ParamodificationOfUnitalNC.
‣ ParamodificationsOfUnitalWithBlock( u, b ) | ( function ) |
Returns: all paramodifications of the unital u with respect to the block b. The results are reduced up to isomorphism of abstract unitals.
‣ AllParamodificationsOfUnital( u ) | ( function ) |
Returns: all paramodifications of the unital u. The results are reduced up to isomorphism of abstract unitals.
gap> InstallGlobalFunction( AllRegularBlockColorings, function( bls, nr_colors, gr ) > local Gamma, complete_subgraphs, graph_of_cliques, colorings, ret, new_blocks, c, c_vec, i, j; gap> # construct the line graph and its cliques > Gamma := Graph( gr, bls, OnSets, > function( x, y ) return x <> y and Intersection( x, y ) = []; gap> end ); gap> complete_subgraphs := CompleteSubgraphs( Gamma, Size(bls)/nr_colors, 1);; gap> complete_subgraphs := Union( List( complete_subgraphs, > x -> Orbit( AutomorphismGroup( Gamma ), x, OnSets ) ) ); gap> Info( InfoParamod, 3, "cliques of the line graph computed..." ); gap> # construct the graph of cliques and its cliques > graph_of_cliques := Graph( Gamma.group, complete_subgraphs, OnSets, > function( x, y ) return x <> y and Intersection( x, y ) = []; gap> end ); gap> colorings := CompleteSubgraphs( graph_of_cliques, nr_colors, 1 ); gap> Info( InfoParamod, 3, Size( colorings ), " block colorings computed..." ); gap> # construct colorings > ret := []; gap> for c in colorings do > c_vec:=0*[1..Size(bls)]; gap> for i in [1..nr_colors] do > for j in VertexNames( graph_of_cliques )[c[i]] do > c_vec[ Position( bls, VertexNames( Gamma )[j] ) ] := i; gap> od; gap> od; gap> Add(ret, Transformation(c_vec)); gap> od; gap> return ret; gap> end );
gap> InstallGlobalFunction( ParamodificationOfUnitalNC, function( u, b, chi ) > local Cb, n_Cb, C_star_b, intact_blks, B_star; gap> Cb := Filtered( BlocksOfUnital( u ), > x -> Size( Intersection( x, b ) ) = 1 ); gap> n_Cb := Length( Cb ); gap> C_star_b := List( [1..n_Cb], > i -> Union( Difference( Cb[i], b ), [ b[i^chi] ] ) ); gap> intact_blks := Difference( BlocksOfUnital( u ), Cb ); gap> B_star := Union( intact_blks, C_star_b ); gap> return AbstractUnitalByDesignBlocks( B_star ); gap> end );
gap> InstallGlobalFunction( ParamodificationOfUnital, function( u, b, chi ) > local Cb; gap> if not b in BlocksOfUnital( u ) then > Error( "argument 2 must be a block of argument 1"); gap> fi; gap> Cb := Filtered( BlocksOfUnital( u ), > x -> Size( Intersection( x, b ) ) = 1 ); gap> Cb := List( Cb, x -> Difference( x, b) ); gap> if not ForAll( Combinations( [1..Size(Cb)], 2 ), p -> > Intersection(Cb{p})=[] or > (p[1]^chi<>p[2]^chi) > ) then Error( "argument 3 is not a proper block coloring" ); gap> fi; gap> return ParamodificationOfUnitalNC( u, b, chi ); gap> end );
gap> InstallGlobalFunction( ParamodificationsOfUnitalWithBlock, > function( u, b ) > local q, Cb, b_stab, new_unitals, all, allchibmod, i, isom_class, colorings; gap> if not b in BlocksOfUnital( u ) then > Error( "argument 2 must be a block of argument 1"); gap> fi; gap> q := Order( u ); gap> Cb := Filtered( BlocksOfUnital( u ), > x -> Size( Intersection( x, b ) ) = 1 ); gap> # Important: keep the order from BlocksOfUnital > Cb := List( Cb, x -> Difference( x, b ) ); gap> # compute all colorings > b_stab := Stabilizer( AutomorphismGroup( u ), b, OnSets ); gap> colorings := AllRegularBlockColorings( Cb, q + 1, b_stab ); gap> Info( InfoParamod, 1, Size( colorings ), " coloring(s) for the given unital-block pair computed..." ); gap> new_unitals := List( colorings, c -> ParamodificationOfUnitalNC( u, b, c ) ); gap> # reduction up to isomorphism > all := [1..Length( new_unitals )]; gap> allchibmod := []; gap> while all <> [] do > i := Remove( all ); gap> isom_class := Filtered( all, x -> Isomorphism( new_unitals[i], > new_unitals[x] ) <> fail ) ; gap> all := Difference( all, isom_class ); gap> Add( allchibmod, new_unitals[i] ); gap> od; gap> return allchibmod; gap> end );
gap> InstallGlobalFunction( AllParamodificationsOfUnital, function( u ) > local blocks, rep_blocks, allchibmods, uus, b; gap> blocks := BlocksOfUnital( u ); gap> rep_blocks := List( Orbits( AutomorphismGroup( u ), blocks, OnSets ), > orb -> Representative( orb ) ); gap> Info( InfoParamod, 2, Size( rep_blocks ), " block representatives for the unital computed..." ); gap> allchibmods := []; gap> for b in rep_blocks do > uus := ParamodificationsOfUnitalWithBlock( u, b ); gap> uus := Filtered( uus, x -> Isomorphism( x, u ) = fail and > ForAll( allchibmods, y -> Isomorphism( y, x ) = fail ) ); gap> Append( allchibmods, uus ); gap> od; gap> return allchibmods; gap> end );
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