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#W refine.gd DifSets Package Dylan Peifer
##
## Functions find all possible difference set/sum preimages of difference
## sums.
##
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##
#F AllRefinedDifferenceSets( <G>, <N>, <difsums> )
##
## <#GAPDoc Label="AllRefinedDifferenceSets">
## <ManSection>
## <Func Name="AllRefinedDifferenceSets" Arg="G, N, difsums"/>
##
## <Description>
## Returns a list of all difference sets that are preimages of difference
## sums contained in the list <A>difsums</A> of difference sums in group
## <A>G</A> mod its normal subgroup <A>N</A>. Difference sums in
## <A>difsums</A> are all assumed to be the same size.
##
## <Example><![CDATA[
## gap> G := SmallGroup(16, 5);;
## gap> N := Subgroup(G, [G.2, G.4]);;
## gap> AllRefinedDifferenceSets(G, N, [[3,1,1,1], [2,2,2,0]]);
## [ [ 1, 3, 2, 8, 4, 15 ], [ 1, 3, 2, 8, 9, 11 ], [ 1, 3, 2, 13, 4, 11 ],
## [ 1, 3, 2, 13, 9, 15 ], [ 1, 3, 6, 8, 4, 11 ], [ 1, 3, 6, 8, 9, 15 ],
## [ 1, 3, 6, 13, 4, 15 ], [ 1, 3, 6, 13, 9, 11 ], [ 1, 5, 2, 6, 4, 15 ],
## [ 1, 5, 2, 6, 9, 11 ], [ 1, 5, 2, 13, 4, 9 ], [ 1, 5, 2, 13, 11, 15 ],
## [ 1, 5, 6, 8, 4, 9 ], [ 1, 5, 6, 8, 11, 15 ], [ 1, 5, 8, 13, 4, 15 ],
## [ 1, 5, 8, 13, 9, 11 ], [ 1, 10, 2, 6, 4, 11 ], [ 1, 10, 2, 6, 9, 15 ],
## [ 1, 10, 2, 8, 4, 9 ], [ 1, 10, 2, 8, 11, 15 ], [ 1, 10, 6, 13, 4, 9 ],
## [ 1, 10, 6, 13, 11, 15 ], [ 1, 10, 8, 13, 4, 11 ], [ 1, 10, 8, 13, 9, 15 ],
## [ 3, 5, 2, 6, 4, 11 ], [ 3, 5, 2, 6, 9, 15 ], [ 3, 5, 2, 8, 4, 9 ],
## [ 3, 5, 2, 8, 11, 15 ], [ 3, 5, 6, 13, 4, 9 ], [ 3, 5, 6, 13, 11, 15 ],
## [ 3, 5, 8, 13, 4, 11 ], [ 3, 5, 8, 13, 9, 15 ], [ 3, 10, 2, 6, 4, 15 ],
## [ 3, 10, 2, 6, 9, 11 ], [ 3, 10, 2, 13, 4, 9 ], [ 3, 10, 2, 13, 11, 15 ],
## [ 3, 10, 6, 8, 4, 9 ], [ 3, 10, 6, 8, 11, 15 ], [ 3, 10, 8, 13, 4, 15 ],
## [ 3, 10, 8, 13, 9, 11 ], [ 5, 10, 2, 8, 4, 15 ], [ 5, 10, 2, 8, 9, 11 ],
## [ 5, 10, 2, 13, 4, 11 ], [ 5, 10, 2, 13, 9, 15 ], [ 5, 10, 6, 8, 4, 11 ],
## [ 5, 10, 6, 8, 9, 15 ], [ 5, 10, 6, 13, 4, 15 ], [ 5, 10, 6, 13, 9, 11 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "AllRefinedDifferenceSets" );
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##
#F NrAllRefinedSets( <G>, <N>, <difsums> )
##
## <#GAPDoc Label="NrAllRefinedSets">
## <ManSection>
## <Func Name="NrAllRefinedSets" Arg="G, N, difsums"/>
##
## <Description>
## Returns the number of preimages that will need to be checked during a
## call to <Ref Func="AllRefinedDifferenceSets"/> with the same arguments.
## This can give a rough estimate of how long the call will take to
## complete.
##
## <Example><![CDATA[
## gap> G := SmallGroup(16, 5);;
## gap> N := Subgroup(G, [G.2, G.4]);;
## gap> NrAllRefinedSets(G, N, [[3,1,1,1], [2,2,2,0]]);
## 472
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "NrAllRefinedSets" );
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##
#F SomeRefinedDifferenceSets( <G>, <N>, <difsums> )
##
## <#GAPDoc Label="SomeRefinedDifferenceSets">
## <ManSection>
## <Func Name="SomeRefinedDifferenceSets" Arg="G, N, difsums"/>
##
## <Description>
## Returns a list of some difference sets that are preimages of difference
## sums contained in the list <A>difsums</A> of difference sums in group
## <A>G</A> mod its normal subgroup <A>N</A>. At least one member of each
## equivalence class that would appear in the set of all preimages will be
## returned, but all preimage difference sets may not appear. Difference
## sums in <A>difsums</A> are all assumed to be the same size. Current
## implementation forces the choice of an identity element when possible.
##
## <Example><![CDATA[
## gap> G := SmallGroup(16, 5);;
## gap> N := Subgroup(G, [G.2, G.4]);;
## gap> SomeRefinedDifferenceSets(G, N, [[3,1,1,1], [2,2,2,0]]);
## [ [ 1, 3, 2, 8, 4, 15 ], [ 1, 3, 2, 8, 9, 11 ], [ 1, 3, 2, 13, 4, 11 ],
## [ 1, 3, 2, 13, 9, 15 ], [ 1, 3, 6, 8, 4, 11 ], [ 1, 3, 6, 8, 9, 15 ],
## [ 1, 3, 6, 13, 4, 15 ], [ 1, 3, 6, 13, 9, 11 ], [ 1, 5, 2, 6, 4, 15 ],
## [ 1, 5, 2, 6, 9, 11 ], [ 1, 5, 2, 13, 4, 9 ], [ 1, 5, 2, 13, 11, 15 ],
## [ 1, 5, 6, 8, 4, 9 ], [ 1, 5, 6, 8, 11, 15 ], [ 1, 5, 8, 13, 4, 15 ],
## [ 1, 5, 8, 13, 9, 11 ], [ 1, 10, 2, 6, 4, 11 ], [ 1, 10, 2, 6, 9, 15 ],
## [ 1, 10, 2, 8, 4, 9 ], [ 1, 10, 2, 8, 11, 15 ], [ 1, 10, 6, 13, 4, 9 ],
## [ 1, 10, 6, 13, 11, 15 ], [ 1, 10, 8, 13, 4, 11 ], [ 1, 10, 8, 13, 9, 15 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "SomeRefinedDifferenceSets" );
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##
#F NrSomeRefinedSets( <G>, <N>, <difsums> )
##
## <#GAPDoc Label="NrSomeRefinedSets">
## <ManSection>
## <Func Name="NrSomeRefinedSets" Arg="G, N, difsums"/>
##
## <Description>
## Returns the number of preimages that will need to be checked during a
## call to <Ref Func="SomeRefinedDifferenceSets"/> with the same arguments.
## This can give a rough estimate of how long the call will take to
## complete.
##
## <Example><![CDATA[
## gap> G := SmallGroup(16, 5);;
## gap> N := Subgroup(G, [G.2, G.4]);;
## gap> NrSomeRefinedSets(G, N, [[3,1,1,1], [2,2,2,0]]);
## 300
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "NrSomeRefinedSets" );
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##
#F IsDifferenceSetByTable( <table>, <D>, <v>, <lambda> )
##
## <#GAPDoc Label="IsDifferenceSetByTable">
## <ManSection>
## <Func Name="IsDifferenceSetByTable" Arg="table, D, v, lambda"/>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "IsDifferenceSetByTable" );
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#F AllRefinedDifferenceSums( <G>, <N1>, <N2>, <difsums> )
##
## <#GAPDoc Label="AllRefinedDifferenceSums">
## <ManSection>
## <Func Name="AllRefinedDifferenceSums" Arg="G, N1, N2, difsums"/>
##
## <Description>
## Returns a list of all difference sums in group <A>G</A> mod its normal
## subgroup <A>N2</A> that are preimages of difference sums contained in the
## list <A>difsums</A> of difference sums in group <A>G</A> mod its normal
## subgroup <A>N1</A>. The subgroup <A>N2</A> must be contained in <A>N1</A>.
## Difference sums in <A>difsums</A> are all assumed to be the same size.
##
## <Example><![CDATA[
## gap> G := SmallGroup(16, 5);;
## gap> N1 := Subgroup(G, [G.2, G.4]);;
## gap> N2 := Subgroup(G, [G.2]);;
## gap> AllRefinedDifferenceSums(G, N1, N2, [[3,1,1,1], [2,2,2,0]]);
## [ [ 1, 1, 0, 1, 0, 1, 2, 0 ], [ 1, 1, 2, 1, 0, 1, 0, 0 ],
## [ 1, 0, 1, 1, 0, 2, 1, 0 ], [ 1, 2, 1, 1, 0, 0, 1, 0 ],
## [ 0, 1, 1, 2, 0, 1, 1, 0 ], [ 2, 1, 1, 0, 0, 1, 1, 0 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "AllRefinedDifferenceSums" );
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#F NrAllRefinedSums( <G>, <N1>, <N2>, <difsums> )
##
## <#GAPDoc Label="NrAllRefinedSums">
## <ManSection>
## <Func Name="NrAllRefinedSums" Arg="G, N1, N2, difsums"/>
##
## <Description>
## Returns the number of preimages that will need to be checked during a
## call to <Ref Func="AllRefinedDifferenceSums"/> with the same arguments.
## This can give a rough estimate of how long the call will take to
## complete.
##
## <Example><![CDATA[
## gap> G := SmallGroup(16, 5);;
## gap> N1 := Subgroup(G, [G.2, G.4]);;
## gap> N2 := Subgroup(G, [G.2]);;
## gap> NrAllRefinedSums(G, N1, N2, [[3,1,1,1], [2,2,2,0]]);
## 22
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "NrAllRefinedSums" );
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#F SomeRefinedDifferenceSums( <G>, <N1>, <N2>, <difsums> )
##
## <#GAPDoc Label="SomeRefinedDifferenceSums">
## <ManSection>
## <Func Name="SomeRefinedDifferenceSums" Arg="G, N1, N2, difsums"/>
##
## <Description>
## Returns a list of some difference sums in group <A>G</A> mod its normal
## subgroup <A>N2</A> that are preimages of difference sums contained in
## the list <A>difsums</A> of difference sums in group <A>G</A> mod its
## normal subgroup <A>N1</A>. At least one member of each equivalence class
## that would appear in the set of all preimages will be returned, but all
## preimage difference sums may not appear. The subgroup <A>N2</A> must be
## contained in <A>N1</A> and difference sums in <A>difsums</A> are all
## assumed to be the same size. Current implementation forces a choice
## of nonzero identity coefficient when possible.
##
## <Example><![CDATA[
## gap> G := SmallGroup(16, 5);;
## gap> N1 := Subgroup(G, [G.2, G.4]);;
## gap> N2 := Subgroup(G, [G.2]);;
## gap> SomeRefinedDifferenceSums(G, N1, N2, [[3,1,1,1], [2,2,2,0]]);
## [ [ 1, 1, 0, 1, 0, 1, 2, 0 ], [ 1, 1, 2, 1, 0, 1, 0, 0 ],
## [ 1, 0, 1, 1, 0, 2, 1, 0 ], [ 1, 2, 1, 1, 0, 0, 1, 0 ],
## [ 2, 1, 1, 0, 0, 1, 1, 0 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "SomeRefinedDifferenceSums" );
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#F NrSomeRefinedSums( <G>, <N1>, <N2>, <difsums> )
##
## <#GAPDoc Label="NrSomeRefinedSums">
## <ManSection>
## <Func Name="NrSomeRefinedSums" Arg="G, N1, N2, difsums"/>
##
## <Description>
## Returns the number of preimages that will need to be checked during a
## call to <Ref Func="SomeRefinedDifferenceSums"/> with the same arguments.
## This can give a rough estimate of how long the call will take to
## complete.
##
## <Example><![CDATA[
## gap> G := SmallGroup(16, 5);;
## gap> N1 := Subgroup(G, [G.2, G.4]);;
## gap> N2 := Subgroup(G, [G.2]);;
## gap> NrSomeRefinedSums(G, N1, N2, [[3,1,1,1], [2,2,2,0]]);
## 21
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "NrSomeRefinedSums" );
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#F DifferenceSumPreImagesOptions( <u>, <w>, <S> )
##
## <#GAPDoc Label="DifferenceSumPreImagesOptions">
## <ManSection>
## <Func Name="DifferenceSumPreImagesOptions" Arg="u, w, S"/>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "DifferenceSumPreImagesOptions" );
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#F DifferenceSumPreImagesPermutations( <opt> )
##
## <#GAPDoc Label="DifferenceSumPreImagesPermutations">
## <ManSection>
## <Func Name="DifferenceSumPreImagesPermutations" Arg="opt"/>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "DifferenceSumPreImagesPermutations" );
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#F DifferenceSumPreImagesPermutationsForced( <opt> )
##
## <#GAPDoc Label="DifferenceSumPreImagesPermutationsForced">
## <ManSection>
## <Func Name="DifferenceSumPreImagesPermutationsForced" Arg="opt"/>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "DifferenceSumPreImagesPermutationsForced" );
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#F DifferenceSumPreImagesTranslate( <cosets>, <u1>, <u2>, <u3>, <perm> )
##
## <#GAPDoc Label="DifferenceSumPreImagesTranslate">
## <ManSection>
## <Func Name="DifferenceSumPreImagesTranslate" Arg="cosets, u1, u2, u3, perm"/>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "DifferenceSumPreImagesTranslate" );
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#F IsDifferenceSumByTable( <table>, <S>, <v>, <k>, <lambda>, <w> )
##
## <#GAPDoc Label="IsDifferenceSumByTable">
## <ManSection>
## <Func Name="IsDifferenceSumByTable" Arg="table, S, v, k, lambda, w"/>
## </ManSection>
## <#/GAPDoc>
DeclareGlobalFunction( "IsDifferenceSumByTable" );
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##
#E
