dual code

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

$$C^{⟂}:=\{d\in {𝔽}_q^n\mid c\cdot d=0\text{ for all }c\in C\}$$

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, $C^{⟂}$ is also a linear code and it is true that if $k$ is the http://planetmath.org/node/5398dimension of $C$, then the of $C^{⟂}$ is $n-k$. It is, however, not necessarily true that $C\cap C^{⟂}=\{0\}$. 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, $(1,1)\in C^{⟂}$. In fact, $C$ equals $C^{⟂}$ in this case. In general, if $C=C^{⟂}$, $C$ is called self-dual. Furthermore $C$ is called self-orthogonal if $C\subseteq C^{⟂}$.

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
Canonical name	DualCode
Date of creation	2013-03-22 15:13:29
Last modified on	2013-03-22 15:13:29
Owner	GrafZahl (9234)
Last modified by	GrafZahl (9234)
Numerical id	6
Author	GrafZahl (9234)
Entry type	Definition
Classification	msc 94B05
Related topic	LinearCode
Related topic	OrthogonalComplement
Defines	self-dual
Defines	self-orthogonal