In coding theory, the dual code of a linear code is the linear code defined by where is a scalar product. In linear algebra terms, the dual code is the annihilator of C with respect to the bilinear form. The dimension of C and its dual always add up to the length n: A generator matrix for the dual code is the parity-check matrix for the original code and vice versa. The dual of the dual code is always the original code.

Self-dual codes

A self-dual code is one which is its own dual. This implies that n is even and dim C = n/2. If a self-dual code is such that each codeword's weight is a multiple of some constant c > 1, then it is of one of the following four types: Codes of types I, II, III, or IV exist only if the length n is a multiple of 2, 8, 4, or 2 respectively. If a self-dual code has a generator matrix of the form G=[I_k|A], then the dual code C^\perp has generator matrix, where I_k is the identity matrix and.