Definition:Multiple of Codeword in Linear Code

Definition

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 $p$.

Examples

Example of Multiple in $V \paren {3, 3}$

In the master code $V \paren {3, 3}$, the codeword $102$ is multiplied by $\eqclass 2 3$ thus:

$2 \paren {102} = 201$

Sources

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