Hamming code

This code is used for single error correction i.e. using this code we can detect only single error. In parity bit method we used only single extra bit but in this method number of extra bits (which also are parity bits) vary with the number of bits of the message.

Suppose we have the number of information bits as m=4 then we have to determine number of parity bits using above relation

2^p >= 4 + p + 1

2^p >= 5 + p

From this we can check for values of p, which one satisfies

For p=1                                 2 >= 6 doesn’t satisfy

For p=2                                 4>= 7 doesn’t satisfy

For p=3                                 8>=8 satisfies hence we have p=3

So now we have 4 information bits and 3 parity bits so total of 7 bits. In the parity bit method, we placed the parity bit at rightmost position. But here we don’t place the extra bits consecutively but the positions are fixed by following rule:

As we need only three positions so we have to pick first 3 which are 1, 2, and 4.

So we have the composition of hamming code as follow:

Bit1        bit2       bit3        bit4        bit5        bit6        bit7

Parity    parity                    parity

P₁                  P₂                  M₁               P₃                  M₂               M₃               M₄

Now we have to decide positions in the hamming code which would be covered by the parity bit i.e. the positions considering which value of parity bit would be decided. We’ll be using following rule for this: