Basic definitions

Enough examples - now for the definition.

Definition 3.3.3   Let $F$ be a finite field. A subset $C$ of $V=F^n$ is called a code of length $n$ . A subspace of $V$ is called a linear code of length $n$ . If $F=\mathbb{F}_2$ then $C$ is called a binary code. If $F=\mathbb{F}_3$ then $C$ is called a ternary code. If $F$ has $q$ elements then $C$ is called a $q$ -ary code. The elements of a code $C$ are called code words or code vectors. Sometimes elements of $F^n$ which do not necessarily belong to $C$ are called ``received vectors''.

The information rate of $C$ is

$$R={\frac{\log_q\vert C\vert}{n}},$$

where $\vert C\vert$ denotes the number of elements of $C$ .