Standard Parity Check Matrix/Examples/(6, 3) code in Z2

Example of Standard Parity Check Matrix

Let $C$ be the linear $\tuple {6, 3}$-code in $\Z_2$ whose standard generator matrix $G$ is given by:

$G := \begin{pmatrix}

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

Its standard parity check matrix $P$ is given by:

$P := \begin{pmatrix}

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

Proof

Expressing $G$ in the form:

$G = \paren {\begin{array} {c|c} \mathbf I_k & \mathbf A \end{array} }$

it is seen that:

$\mathbf A = \begin{pmatrix}

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

It is noted that $\mathbf A^\intercal$ is:

$\mathbf A^\intercal = \begin{pmatrix}

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

as $\mathbf A$ is symmetrical about the main diagonal.

Then each of the elements of $\Z_2$ is self-inverse, so:

$-\mathbf A^\intercal = \mathbf A^\intercal$

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes: Example $6.18$