# Hamming Code, Parity Matrix

#### Niall101

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
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

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

• Niall101