Standard Generator Matrix for Linear Code/Examples/(3, 2) code in Z2

Example of Standard Generator Matrix for Linear Code
Let $G$ be the standard generator matrix:


 * $G := \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$

$G$ generates the linear code $C$:


 * $C = \set {000, 101, 011, 110}$

The minimum distance of $C$ is $2$, so $C$ detects $1$ transmission error and corrects none.

Proof
Multiplying $G$ by the $4$ vectors $00, 01, 10, 11$ in turn gives:

all arithmetic being modulo $2$.

The rest of the result follows by inspection.