Standard Parity Check Matrix/Examples/(7, 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 & 1 \\ 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 1 \end{pmatrix}$

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


 * $P := \begin{pmatrix}

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

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


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

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

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

Then each of the elements of $\Z_2$ is self-inverse, so:
 * $-\mathbf A^\intercal = \mathbf A^\intercal$

Its Syndrome Decoding table is:


 * $\begin{array} {cc}

0000000 & 0000 \\ 1000000 & 1101 \\ 0100000 & 1110 \\ 0010000 & 1011 \\ 0001000 & 1000 \\ 0000100 & 0100 \\ 0000010 & 0010 \\ 0000001 & 0001 \\ 0000011 & 0011 \\ 0000101 & 0101 \\ 0000110 & 0110 \\ 0000111 & 0111 \\ 0001001 & 1001 \\ 0001010 & 1010 \\ 0001100 & 1100 \\ 0001111 & 1111 \\ \end{array}$

The syndrome of $1100011$ is $1101 + 1110 + 0011 = 0000$, so $1100011$ is a codeword of $C$.

The syndrome of $1011000$ is $1101 + 1011 + 1000 = 1110$, so $1011000$ corrects to $1011000 - 0100000 = 1111000$.

The syndrome of $0101110$ is $1110 + 1100 + 0010 = 0000$, so $0101110$ is a codeword of $C$.