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 $p$.