Parity-Check Matrix

Parity-Check Matrix is used to check the parity of a #linear code.

$$ \begin{align} H &= [I_{n-k} \vdots P^T]\\ HG^T &= [I_{n-k} \vdots P^T] \left[ \begin{array}{c} P^T \\ \cdots \\ I_k \end{array} \right]\\ &= P^T + P^T (\mod 2)\\ &= 0\\ HG^T &= GH^T \end{align} $$

Where:

$H$ is the $n-k$ by $n$ Parity-Check Matrix
$I_{n-k}$ is the $n-k$ by $n-k$ Identity Matrix#
$T$ denotes the transpose version of the matrix
$H^T$ is the Syndrome#
$G$ is Generator Matrix

In the Cyclic Code world, Parity-Check Matrix could be constructed using the following formula from #Parity-Check Polynomial:

$$ H = D^k h(D^{-1}), D^{k+1} h(D^{-1}), \ldots, D^{n-1} h(D^{-1}) $$

Links to this page

TIT3131 Chapter 4: Cyclic Code

Parity-Check Matrix
TIT3131 Chapter 3: Linear Block Coding

Parity-Check Matrix
Systematic Code

For example, if the resulting Generator Matrix or Parity-Check Matrix is not systematic, we could XOR the non-systematic part out with rows that exhibits systematic characteristic.
Syndrome Decoding

Syndrome Decoding is a #decoding technique to correct some errors, especially those with high probability, cause by noisy channel. It does this by comparing $r$, a received vector of dimensions $1 \times n$, to #$H$, the parity check matrix. $r$ is defined as:
Parity-Check Polynomial

Parity-Check Polynomial $h(D)$ with a degree $k$ can uniquely specify a $(n, k)$ #Cyclic Code equivalent to Parity-Check Matrix#. It is defined as follow:
Linear Block Coding

The parity bits will be checked against on the receiver side using the mathematical relationship of the parity bit and message bit represented by Parity-Check Matrix#.

#math #encoding