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:

considering the rows of $G$ as codewords
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 $u \in U$.