65.99 DualCode

DualCode( C )

DualCode returns the dual code of C. The dual code consists of all codewords that are orthogonal to the codewords of C. If C is a linear code with generator matrix G, the dual code has parity check matrix G (or if C has parity check matrix H, the dual code has generator matrix H). So if C is a linear [n, k] code, the dual code of C is a linear [n, n-k] code. If C is a cyclic code with generator polynomial g(x), the dual code has the reciprocal polynomial of g(x) as check polynomial.

The dual code is always a linear code, even if C is non-linear.

If a code C is equal to its dual code, it is called self-dual.

    gap> R := ReedMullerCode( 1, 3 );
    a linear [8,4,4]2 Reed-Muller (1,3) code over GF(2)
    gap> RD := DualCode( R );
    a linear [8,4,4]2 Reed-Muller (1,3) code over GF(2)
    gap> R = RD;
    true
    gap> N := WholeSpaceCode( 7, GF(4) );
    a cyclic [7,7,1]0 whole space code over GF(4)
    gap> DualCode( N ) = NullCode( 7, GF(4) );
    true 

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