Parity checks as matrix multiplication

A partity check can be represented as a form of matrix multiplication.

Imagine we have an input bit stream, a, of length k:

Then a matrix of the form below will perform an even parity check if k is odd (or an odd parity check if k is even):

Here’s a very simple worked example:

Let a = 10110

Parity check matrix =

Output = 101101 (modulo 2 arithmetic is used to compute each ‘column’ in the output)

Of course parity checks are a crude form of 1 bit error detection and do not allow for error correction.