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 $2 \ \text{(c)}$