Standard Generator Matrix for Linear Code/Examples/(5, 3) code in Z5
Example of Standard Generator Matrix for Linear Code
Let $G$ be the standard generator matrix over $\Z_5$:
- $G := \begin{pmatrix}
1 & 0 & 0 & 2 & 1 \\ 0 & 1 & 0 & 1 & 3 \\ 0 & 0 & 1 & 4 & 1 \\ \end{pmatrix}$
$G$ generates a linear code which detects $1$ transmission error and corrects $0$ transmission errors.
Proof
Let $C$ denote the linear code generated by $G$.
There are $5^3 = 125$ codewords in $C$, so it is impractical to list them all.
We have that:
\(\ds 2 \times 1 0 0 2 1 + 0 1 0 1 3\) | \(=\) | \(\ds 2 0 0 4 2 + 0 1 0 1 3\) | \(\ds \pmod 5\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 1 0 0 0\) | \(\ds \pmod 5\) |
so $C$ has a minimum distance of at least $2$.
Looking at the first $3$ columns shows there can be no codeword with weight less than $2$.
From Minimum Distance of Linear Code is Smallest Weight of Non-Zero Codeword, the minimum distance of $C$ is $2$.
From Error Detection Capability of Linear Code, $C$ can detect $2 - 1 = 1$ transmission errors.
From Error Correction Capability of Linear Code, $C$ can correct $\floor {\dfrac {2 - 1} 2} = 0$ transmission errors.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes: Exercise $2 \ \text{(c)}$