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 thestandard dot product or the Hermitian dot product.

This definition is reminiscent of orthogonal complements of http://planetmath.org/node/5398finitedimensional vector spaces over the real or complex numbers. Indeed,$C^{⟂}$ is also a linear code and it is true that if $k$ is thehttp://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 blocklength $2$ 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 thiscase. In general, if $C=C^{⟂}$, $C$ is calledself-dual. Furthermore $C$ is called self-orthogonal if$C\subseteq C^{⟂}$.

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