Definition:Linear Code

Definition
Let $p$ be a prime number.

Let $\Z_p$ be the set of residue classes modulo $p$.

Let $\map V {n, p}$ denote the set of sequences of length $n$ of elements of $\Z_p$.

A linear $\tuple {n, k}$-code is a $k$-dimensional subspace $C$ of $\map V {n, p}$ considered as a vector space over $\Z_p$ of $n$ dimensions.

Codeword
An element of $\map V {n, p}$ is usually written without punctuation, so that, for example:
 * $\paren {\eqclass 1 p, \eqclass 0 p, \eqclass 1 p}$

is presented as:
 * $101$

Also see

 * Set of Sequences of Residue Classes forms Vector Space: $\map V {n, p}$ is a vector space over $\Z_p$ of $n$ dimensions