Generation of Linear Code from Standard Generator Matrix
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Theorem
Let $G$ be a (standard) generator matrix for a linear code.
The following methods can be used to generate a linear code from $G$:
Method 1
A linear code $C$ can be obtained from $G$ by:
- forming all possible linear combinations of those codewords, considering them as vectors of a vector space.
Method 2
A linear code $C$ can be obtained from $G$ by:
- taking the set $U$ of all sequences of length $k$ over $\Z_p$ and expressing them as $1 \times k$ matrices
- forming all possible matrix products $u G$ for all $u \in U$.