Generation of Linear Code from Standard Generator Matrix/Method 1
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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:
- forming all possible linear combinations of those codewords, considering them as vectors of a vector space.
Proof
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Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes: Definition $6.11$