\begin{definition}[Definition:Multiple of Codeword in Linear Code]
Let $\map V {n, p}$ denote the linear $\tuple {n, n}$-code modulo $p$.
The operation of '''multiplication''' on $\map V {n, p}$ is defined as follows.
Let $A$ be an elements of $\map V {n, p}$, that is, a sequence of length $n$ of residue classes modulo $p$.
Let $a_k$ denote the $k$th term of $A$.
Let $c \in \Z_p$ be an element of the set $\Z_p$ of residue classes modulo $p$.
Then $c A$ is the sequence of length $n$ of residue classes modulo $p$ whose $k$th term $c_k$ is defined as:
:$c_k : = c \times_p a_k$
where $\times_p$ denotes the operation of multiplication modulo $p$.
\end{definition}