Definition:Standard Parity Check Matrix

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 $p$.

$G := \begin{pmatrix}

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

$P := \begin{pmatrix}

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

$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}$

$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}$

1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes: Definition $6.17$