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:
 * 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 $$\bar{a}=a^q\in\mathbb{F}_q$$.