Generation of Linear Code from Standard Generator Matrix/Method 2

Theorem

Let $G$ be a (standard) $k \times n$ generator matrix over $\Z_p$ for a linear code.

The following method can be used to generate from $G$ a linear $\tuple {n, k}$ code over $\Z_p$:

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 $u \in U$.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes: Definition $6.11$