Automorphism Groups

PermutationGroupHadamardCodeZ4(δ, m): RngIntElt, RngIntElt -> GrpPerm, Mtrx

PAutHadamardCodeZ4(δ, m): RngIntElt, RngIntElt -> GrpPerm, Mtrx

Given an integer \(m\geq 1\) and an integer \(\delta\) such that \(1\leq \delta \leq \lfloor (m+1)/2 \rfloor\), this function returns the permutation group \(G\) of a Hadamard code over \({\mathbb{Z}}_4\) of length \(2^{m-1}\) and type \(2^\gamma 4^\delta\), where \(\gamma=m+1-2\delta\). The group \(G\) contains all permutations of the coordinates which preserve the code. Thus only permutation of coordinates is allowed, and the degree of \(G\) is always \(2^{m-1}\). Moreover, the generator matrix with \(\gamma+\delta\) rows used to generate the code is returned. This matrix is constructed in a recursive way using the Plotkin and BQPlotkin constructions defined in Section New Codes from Old.

PermutationGroupHadamardCodeZ4Order(δ, m): RngIntElt, RngIntElt -> RngIntElt

PAutHadamardCodeZ4Order(δ, m): RngIntElt, RngIntElt -> RngIntElt

Given an integer \(m\geq 1\) and an integer \(\delta\) such that \(1\leq \delta \leq \lfloor (m+1)/2 \rfloor\), return the order of the permutation group \(G\) of a Hadamard code over \({\mathbb{Z}}_4\) of length \(2^{m-1}\) and type \(2^\gamma 4^\delta\), where \(\gamma=m+1-2\delta\). The group \(G\) contains all permutations of the coordinates which preserve the code.

PermutationGroupExtendedPerfectCodeZ4(δ, m): RngIntElt, RngIntElt -> GrpPerm, Mtrx

PAutExtendedPerfectCodeZ4(δ, m): RngIntElt, RngIntElt -> GrpPerm, Mtrx

Given an integer \(m\geq 2\) and an integer \(\delta\) such that \(1\leq \delta \leq \lfloor (m+1)/2 \rfloor\), return the permutation group \(G\) of an extended perfect code over \({\mathbb{Z}}_4\) of length \(2^{m-1}\), such that its dual code is of type \(2^\gamma 4^\delta\), where \(\gamma=m+1-2\delta\). The group \(G\) contains all permutations of the coordinates which preserve the code. Thus only permutation of coordinates is allowed, and the degree of \(G\) is always \(2^{m-1}\). Moreover, the generator matrix with \(\gamma+\delta\) rows used to generate the code is returned. This matrix is constructed in a recursive way using the Plotkin and BQPlotkin constructions defined in Section New Codes from Old.

PermutationGroupExtendedPerfectCodeZ4Order(δ, m): RngIntElt, RngIntElt -> RngIntElt

PAutExtendedPerfectCodeZ4Order(δ, m): RngIntElt, RngIntElt -> RngIntElt

Given an integer \(m\geq 2\) and an integer \(\delta\) such that \(1\leq \delta \leq \lfloor (m+1)/2 \rfloor\), return the order of the permutation group \(G\) of an extended perfect code over \({\mathbb{Z}}_4\) of length \(2^{m-1}\), such that its dual code is of type \(2^\gamma 4^\delta\), where \(\gamma=m+1-2\delta\). The group \(G\) contains all permutations of the coordinates which preserve the code.

Example: Spain Z4 8 (ex-61ad8c)

> C := HadamardCodeZ4(2,4);
> PAut := PAutHadamardCodeZ4(2,4);
> PAut;
Permutation group PAut acting on a set of cardinality 8
    (1, 2)(3, 4)(5, 6)(7, 8)
    (2, 4)(6, 8)
    (5, 7)(6, 8)
    (1, 5)(3, 7)
> {p : p in Sym(8) | C^p eq C} eq Set(PAut);
true
> #PAut eq PAutHadamardCodeZ4Order(2,4);
true
> d := 2; m := 4; g := m+1-2*d;
> PAutHadamardCodeZ4Order(d, m) eq
>  #GL(d-1,Integers(4))*#GL(g,Integers(2))*2^g*4^((g+1)*(d-1));
true
> d := 4; m := 8; g := m+1-2*d;
> PAutHadamardCodeZ4Order(d, m) eq
>   #GL(d-1,Integers(4))*#GL(g,Integers(2))*2^g*4^((g+1)*(d-1));
true
> PAutHadamardCodeZ4(2,4) eq PAutExtendedPerfectCodeZ4(2,4);
true

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PermutationGroup(C): CodeLinRng -> GrpPerm

The permutation group \(G\) of the linear code \(C\) of length \(n\) over the ring \(R\), where \(G\) is the group of all permutation-action permutations which preserve the code. Thus only permutation of coordinates is allowed, and the degree of \(G\) is always \(n\).

PermutationGroupGrayMapImage(C): CodeLinRng -> GrpPerm

Given a code \(C\) over \({\mathbb{Z}}_4\) of length \(n\), return the permutation group \(G_{bin}\) of \(C_{bin}=\Phi(C)\), where \(G_{bin}\) is the group of all permutation-action permutations which preserve the binary code \(C_{bin}\) of length \(2n\) and \(\Phi\) is the Gray map. Thus only permutation of coordinates is allowed, and the degree of \(G_{bin}\) is always \(2n\).

Example: Code Z4 Perm Group (ex-326517)

> C := HadamardCodeZ4(2,4);
> PAutC := PermutationGroup(C);
> PAutC eq PAutHadamardCodeZ4(2,4);
true
> #PAutC eq PAutHadamardCodeZ4Order(2,4);
true
> {p : p in Sym(8) | C^p eq C} eq Set(PAutC);
true

> Cbin := GrayMapImage(C);
> PAutCbin := PermutationGroupGrayMapImage(C);
> {p : p in PAutCbin | Set(Cbin)^p eq Set(Cbin)} eq Set(PAutCbin);
true

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