# Conditions Satisfied by Linear Code

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## 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 $6.2$