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

Example of Standard Generator Matrix for Linear Code

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

$G$ 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.

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

$\ds \begin{pmatrix} 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$

$=$

$\ds \begin{pmatrix} 0 \times 1 + 0 \times 0 & 0 \times 0 + 0 \times 1 & 0 \times 1 + 0 \times 1 \end{pmatrix}$

$\ds $

$=$

$\ds \begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$

$\ds \begin{pmatrix} 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$

$=$

$\ds \begin{pmatrix} 0 \times 1 + 1 \times 0 & 0 \times 0 + 1 \times 1 & 0 \times 1 + 1 \times 1 \end{pmatrix}$

$\ds $

$=$

$\ds \begin{pmatrix} 0 & 1 & 1 \end{pmatrix}$

$\ds \begin{pmatrix} 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$

$=$

$\ds \begin{pmatrix} 1 \times 1 + 0 \times 0 & 1 \times 0 + 0 \times 1 & 1 \times 1 + 0 \times 1 \end{pmatrix}$

$\ds $

$=$

$\ds \begin{pmatrix} 1 & 0 & 1 \end{pmatrix}$

$\ds \begin{pmatrix} 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$

$=$

$\ds \begin{pmatrix} 1 \times 1 + 1 \times 0 & 1 \times 0 + 1 \times 1 & 1 \times 1 + 1 \times 1 \end{pmatrix}$

$\ds $

$=$

$\ds \begin{pmatrix} 1 & 1 & 0 \end{pmatrix}$

all arithmetic being modulo $2$.

The rest of the result follows by inspection.

$\blacksquare$

1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes: Example $6.12$