Dual code

In coding theory, the dual code of a linear code

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

is the linear code defined by

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

where

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

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

${\displaystyle \dim C+\dim C^{\perp }=n.}$

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.