Dual code

In coding theory, the dual code of a linear code

C\subset \mathbb {F} _{q}^{n}

is the linear code defined by

C^{\perp }=\{x\in \mathbb {F} _{q}^{n}\mid \langle x,c\rangle =0\;\forall c\in C\}

where

\langle x,c\rangle =\sum _{i=1}^{n}x_{i}c_{i}

is a scalar product. In linear algebra terms, the dual code is the annihilator of C with respect to the bilinear form $\langle \cdot \rangle$ . The dimension of C and its dual always add up to the length n:

\dim C+\dim C^{\perp }=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:^[1]

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.

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 $[-{\bar {A}}^{T}|I_{k}]$ , where $I_{k}$ is the $(n/2)\times (n/2)$ identity matrix and ${\bar {a}}=a^{q}\in \mathbb {F} _{q}$ .