|
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 a 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 , then it is of one of the following four types: *Type I codes are binary self-dual codes which are not doubly even. Type I codes are always even (every codeword has even Hamming weight). *Type II codes are binary self-dual codes which are doubly even. *Type III codes are ternary self-dual codes. Every codeword in a Type III code has Hamming weight divisible by 3. *Type IV codes are self-dual codes over F4. These are again even. Codes of types I, II, III, or IV exist only if the length ''n'' is a multiple of 2, 8, 4, or 2 respectively. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「dual code」の詳細全文を読む スポンサード リンク
|