Hamming Code, Parity Matrix

An Hamming (n,k) code is the set of all n-simbols binary word that satisfy the relation...

$\overline{z}= \overline {x} \cdot H= \overline {0}$ (1)

... where...

a) $\overline {x}$ is a n bit row vector

b) $\overline {z}$ is a n-k bit column vector called symdrome

c) $\overline {0}$ is the n-k bit 'null vector'

d) $H$ is the n x k bit 'parity check' matrix

An example of parity 7 x 4 Hamming parity check matrix is the following...

$H = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 1 & 0\\ 0 & 1 & 1\\ 1 & 1 & 1\\ 1 & 0 & 1 \end{bmatrix}$

A 7 x 4 Hamming code can correct any single error that occurred in a 7 bits transmitted word...

Kind regards

$\chi$ $\sigma$