Let $C$ be a linear code of block length $n$ over the finite field $\mbb{F}_q$ Then the set \begin{equation*} C^\perp:=\{d\in\mbb{F}_q^n\mid c\cdot d=0\text{ for all }c\in C\} \end{equation*}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 finite 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 dimension of $C$ then the dimension 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 $2$ spanned 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 $24$