‣ CodeWordByGroupRingElement( F, S, a ) | ( operation ) |
Returns: The code word of length the length of S associated to the group ring element a.
The input F should be a finite field. The input S is a fixed ordering of a group \(G\) and a is an element in the group algebra \(FG\).
Each element \(c\) in \(FG\) is of the form \( c=\sum_{i=1}^n f_i g_i\), where we fix an ordering \(\{g_1,g_2,...,g_n \}\) of the group elements of \(G\) and \(f_i\in F\). If we look at \(c\) as a codeword, we will write \([f_1 f_2 ... f_n]\). (9.23).
gap> G:=DihedralGroup(8);; gap> F:=GF(3);; gap> FG:=GroupRing(F,G);; gap> a:=AsList(FG)[27]; (Z(3)^0)*<identity> of ...+(Z(3)^0)*f1+(Z(3)^0)*f2+(Z(3)^0)*f3+(Z(3)^ 0)*f1*f2+(Z(3)^0)*f2*f3+(Z(3))*f1*f2*f3 gap> S:=AsSet(G); [ <identity> of ..., f1, f2, f3, f1*f2, f1*f3, f2*f3, f1*f2*f3 ] gap> CodeWordByGroupRingElement(F,S,a); [ Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3), Z(3)^0, Z(3) ]
‣ CodeByLeftIdeal( F, G, S, I ) | ( operation ) |
Returns: All code words of length the length of S associated to the group ring elements in the ideal I of FG.
The input F should be a finite field. The input S is a fixed ordering of a group \(G\) and I is a left ideal of the group algebra \(FG\).
Each element \(c\) in \(FG\) is of the form \( c=\sum_{i=1}^n f_i g_i\), where we fix an ordering \(\{g_1,g_2,...,g_n \}\) of the group elements of \(G\) and \(f_i\in F\). If we look at \(c\) as a codeword, we will write \([f_1 f_2 ... f_n]\). (9.23).
gap> G:=DihedralGroup(8);; gap> F:=GF(3);; gap> FG:=GroupRing(F,G);; gap> S:=AsSet(G); [ <identity> of ..., f1, f2, f3, f1*f2, f1*f3, f2*f3, f1*f2*f3 ] gap> H:=StrongShodaPairs(G)[5][1]; Group([ f1*f2*f3, f3 ]) gap> K:=StrongShodaPairs(G)[5][2]; Group([ f1*f2 ]) gap> N:=Normalizer(G,K); Group([ f1*f2*f3, f3 ]) gap> epi:=NaturalHomomorphismByNormalSubgroup(N,K); [ f1*f2*f3, f3 ] -> [ f1, f1 ] gap> QHK:=Image(epi,H); Group([ f1, f1 ]) gap> gq:=MinimalGeneratingSet(QHK)[1]; f1 gap> C:=CyclotomicClasses(Size(F),Index(H,K))[2]; [ 1 ] gap> e:=PrimitiveIdempotentsNilpotent(FG,H,K,C,[epi,gq]); [ (Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3)^0)*f1*f2+(Z(3))*f1*f2*f3, (Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3))*f1*f2+(Z(3)^0)*f1*f2*f3 ] gap> FGe := LeftIdealByGenerators(FG,[e[1]]);; gap> V := VectorSpace(F,CodeByLeftIdeal(F,G,S,FGe));; gap> B := Basis(V);; gap> LoadPackage("guava");; gap> code := GeneratorMatCode(B,F); a linear [8,2,1..4]4..5 code defined by generator matrix over GF(3) gap> MinimumDistance(code); 4
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