Hamming Code, Parity Matrix

Aug 2008
98
8
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!
 

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chisigma

MHF Hall of Honor
Mar 2009
2,162
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near Piacenza (Italy)
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

\(\displaystyle \chi\) \(\displaystyle \sigma\)
 
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