Conditions Satisfied by Linear Code

Theorem

Let $p$ be a prime number.

Let $\Z_p$ be the set of residue classes modulo $p$.

Let $C := \tuple {n, k}$ be a linear code of a master code $\map V {n, p}$.

Then $C$ satisfies the following conditions:

$(C \, 1): \quad \forall \mathbf x, \mathbf y \in C: \mathbf x + \paren {-\mathbf y} \in C$

$(C \, 2): \quad \forall \mathbf x \in C, m \in \Z_p: m \times \mathbf x \in C$

where $+$ and $\times$ are the operations of codeword addition and codeword multiplication respectively.

This article is complete as far as it goes, but it could do with expansion. In particular: Add a page defining the difference between codewords. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. 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 {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Proof

From Master Code forms Vector Space, $\map V {n, p}$ is a vector space.

By definition, $\tuple {n, k}$ is a subspace of $\map V {n, p}$.

The result follows by the fact that a subspace is itself a vector space.

This needs considerable tedious hard slog to complete it. In particular: I lose patience with the fine detail. 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 {{Finish}} 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 $6.2$