Information Theory and Coding > Error Control Coding > How to check whether there is any errors in receiving message?

How to check whether there is any errors in receiving message?

Suppose we have received codeword c in which first k bits are message and last q bits are check bits.  To check the received message we need a matrix called parity check matrix as follows:

For a received codeword the verification of correction is obtained by multiplying the code with H^T.

cH^T = 0

if it is zero matrix then there is no error and if 1 appears then error is in that position. Let the position is r = 5 then the error vector is

E = [0 0 0 0 1 0]

 

So the corrected codeword will be cꚚE

What is Parity Check Matrix?

A parity check matrix provides a simple method of detecting whether an error has occurred or not. If the multiplication of the received word with the transpose of H yields a non-zero vector, it implies that an error has occurred.

cH^T = 0

The size of the parity check matrix is ( n - k) x n  i.e. q x n

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