Definition:Standard Parity Check Matrix

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Definition

Let $C$ be a linear $\tuple {n, k}$ code whose master code is $\map V {n, p}$.

Let $G$ be the $k \times n$ standard generator matrix of $C$:

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

where:

$\mathbf I$ denotes the identity matrix of order $k$
$\mathbf A$ denotes some $k \times \paren {n - k}$ matrix.


The (standard) parity check matrix associated with $G$ is the $\paren {n - k} \times n$ matrix:

$P = \paren {\begin{array} {c|c} -\mathbf A^\intercal & \mathbf I_{n - k} \end{array} }$

where:

$\mathbf A^\intercal$ denotes the transpose of $\mathbf A$
$\mathbf I_{n - k}$ denotes the identity matrix of order $n - k$
the $-$ sign before $\mathbf A^\intercal$ denotes that each of the elements of $\mathbf A$ is to be replaced with its inverse element in $\Z_p$, the additive group of integers modulo $p$.


Examples

Linear $\tuple {6, 3}$-code in $\Z_2$

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:

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


Linear $\tuple {7, 3}$-code in $\Z_2$

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:

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


Sources