Definition:Addition of Codewords in Linear Code

Definition
Let $\map V {n, p}$ denote the linear $\tuple {n, n}$-code modulo $p$.

The operation of addition on $\map V {n, p}$ is defined as follows.

Let $A$ and $B$ be elements of $\map V {n, p}$, that is, sequences of length $n$ of residue classes modulo $p$.

Let $a_k$ and $b_k$ denote the $k$th term of $A$ and $B$ respectively.

Then $C = A + B$ is the sequence of length $n$ of residue classes modulo $p$ whose $k$th term $c_k$ is defined as:
 * $c_k : = a_k +_p b_k$

where $+_p$ denotes the operation of addition modulo $p$.