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 $p$.

Examples

Example of Addition in $V \paren {3, 2}$

In the master code $V \paren {3, 2}$, the codewords $110$ and $011$ are added thus:

$110 + 011 = 101$

Example of Addition in $V \paren {2, 3}$

In the master code $V \paren {2, 3}$, the codewords $12$ and $11$ are added thus:

$12 + 11 = 20$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes