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:

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.