65.42 AutomorphismGroup

AutomorphismGroup( C )

AutomorphismGroup returns the automorphism group of a binary code C. This is the largest permutation group of degree n such that each permutation applied to the columns of C again yields C. GUAVA uses the external program desauto from J.S. Leon to compute the automorphism group. The function PermutedCode permutes the columns of a code (see PermutedCode).

    gap> R := RepetitionCode(7,GF(2));
    a cyclic [7,1,7]3 repetition code over GF(2)
    gap> AutomorphismGroup(R);
    Group( (1,7), (2,7), (3,7), (4,7), (5,7), (6,7) )
                            # every permutation keeps R identical
    gap> C := CordaroWagnerCode(7);
    a linear [7,2,4]3 Cordaro-Wagner code over GF(2)
    gap> Elements(C);
    [ [ 0 0 0 0 0 0 0 ], [ 1 1 1 1 1 0 0 ], [ 0 0 1 1 1 1 1 ],
      [ 1 1 0 0 0 1 1 ] ]
    gap> AutomorphismGroup(C);
    Group( (3,4), (4,5), (1,6)(2,7), (1,2), (6,7) )
    gap> C2 :=  PermutedCode(C, (1,6)(2,7));
    a linear [7,2,4]3 permuted code
    gap> Elements(C2);
    [ [ 0 0 0 0 0 0 0 ], [ 0 0 1 1 1 1 1 ], [ 1 1 1 1 1 0 0 ],
      [ 1 1 0 0 0 1 1 ] ]
    gap> C2 = C;
    true 

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