dual code

Let $C$ be a linear code of block length $n$ over the finite field $\mathbb{F}_{q}$. Then the set

 $C^{\perp}:=\{d\in\mathbb{F}_{q}^{n}\mid c\cdot d=0\text{ for all }c\in C\}$

is the dual code of $C$. Here, $c\cdot d$ denotes either the standard dot product or the Hermitian dot product.

This definition is reminiscent of orthogonal complements of http://planetmath.org/node/5398finite dimensional vector spaces over the real or complex numbers. Indeed, $C^{\perp}$ is also a linear code and it is true that if $k$ is the http://planetmath.org/node/5398dimension of $C$, then the of $C^{\perp}$ is $n-k$. It is, however, not necessarily true that $C\cap C^{\perp}=\{0\}$. For example, if $C$ is the binary code of block length $2$ http://planetmath.org/node/806spanned by the codeword $(1,1)$ then $(1,1)\cdot(1,1)=0$, that is, $(1,1)\in C^{\perp}$. In fact, $C$ equals $C^{\perp}$ in this case. In general, if $C=C^{\perp}$, $C$ is called self-dual. Furthermore $C$ is called self-orthogonal if $C\subseteq C^{\perp}$.

Famous examples of self-dual codes are the extended binary Hamming code of block length $8$ and the extended binary Golay code of block length $24$.

Title dual code DualCode 2013-03-22 15:13:29 2013-03-22 15:13:29 GrafZahl (9234) GrafZahl (9234) 6 GrafZahl (9234) Definition msc 94B05 LinearCode OrthogonalComplement self-dual self-orthogonal