Hamming Code, Parity Matrix

Have an exam tomorrow. Pretty sure a question like this will be on it. Am a bit lost on this one.

Any help greatly appreciated. Thanks!

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

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

... where...

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

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

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

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

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

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$