65.101 CosetCode

CosetCode( C, w )

CosetCode returns the coset of a code C with respect to word w. w must be of the codeword type. Then, w is added to each codeword of C, yielding the elements of the new code. If C is linear and w is an element of C, the new code is equal to C, otherwise the new code is an unrestricted code.

Generating a coset is also possible by simply adding the word w to C. See Operations for Codes.

    gap> H := HammingCode(3, GF(2));
    a linear [7,4,3]1 Hamming (3,2) code over GF(2)
    gap> c := Codeword("1011011");; c in H;
    false
    gap> C := CosetCode(H, c);
    a (7,16,3)1 coset code
    gap> List(Elements(C), el-> Syndrome(H, el));
    [ [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ],
      [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ],
      [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ] ]
    # All elements of the coset have the same syndrome in H 

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