Definition:Multiple of Codeword in Linear Code
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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