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