Dual code

In coding theory, the dual code of a linear code

$C\subset \mathbb {F} _{2}^{n}$

is the linear code defined by

$C^{\perp }=\{x\in \mathbb {F} _{2}^{n}\mid <x,c>=0\forall c\in C\}$

Here <,> denotes the vector dot product, which is taken over the field $\mathbb {F} _{2}$. In simpler language, it consists of all code words, as binary strings, that have 1s in places overlapping the 1s in each word from C always at an even number of locations.

In linear algebra terms, the dual code is the annihilator of C with respect to the bilinear form <,>. An important property is that the dual of the dual code is the original code itself. This follows from the fact that the dimensions of C and its dual always add up to n.