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### 8 Applications of the Wedderga package

#### 8.1 Coding theory applications

##### 8.1-1 CodeWordByGroupRingElement
 ‣ 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) ]



##### 8.1-2 CodeByLeftIdeal
 ‣ 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);;