Example 2: Hamming code and Parity bit

Eg.  Now form a hamming code for 5-bit information bits 10110 with odd parity

m=5 and we have to follow         2^p >= m + p + 1

The value of p as 4 to satisfy 2⁴ (16) >= 5 + 4 + 1 but p=3 doesn’t satisfy as 2³ (8) >= 5 + 3 + 1

So p=4 and hence a total of 9 bits

Parity bit positions are 1, 2, 4, 8

And hence hamming code composition is as follow

0001       0010      0011      0100       0101       0110       0111       1000       1001 

1              2              3              4              5              6              7              8              9

P₁                P₂             M₁             P₃              M₂            M₃               M₄               P₄             M₅

Now if we see P₁ has 1 at LSB so message bits with this parity bit are M₁ M₂ M₄ M₅

Similarly we see P₂ checks M₁ M₃ M₄

Similarly we see P₃ checks M₂ M₃ M₄

Similarly we see P₄ checks M₅

Or we can also put it as

P₁ checks code bits 1, 3, 5, 7, 9

P₂ checks code bits 2, 3, 6, 7

P₃ checks code bits 4, 5, 6, 7

P₄ checks code bits 8, 9

For message 10110 we hamming code

Bit positions  

1              2              3              4              5              6              7              8              9 

P₁               P₂               1 P₃             0 1 1 P₄              0

We see P₁=1 to make no. of 1’s to 3 i.e. odd

We see P₂=0 to make no. of 1’s to 3 i.e. odd

We see P₃=1 to make no. of 1’s to 3 i.e. odd

We see P₄=1 to make no. of 1’s to 1 i.e. odd

So we get the hamming code as 101101110