Question

Question 2. Using the Hamming code algorithm (7,4), convert a data message (0111) using 7bit. a) b) c) d) e) Find number of parity bits needed Evaluate values of parity bits using Exclusive-OR. Show final message bits with parity bits. How do you identify that the received message has error? Inject an error (o or 1) at position 3 and identify the error position.

Answer 1

given message is 0111

So here

m₁ = 0

m₂ = 1

m₃ = 1

m₄ = 1

a).

Number of parity bits = Total bits - message bits = 7- 4 = 3

b).

1	2	3	4	5	6	7
P1	P2	m1	P3	m2	m3	m4
P1	P2	0	P3	1	1	1

Now we will calculate the even parity using Exclusive - OR

P₁ --> 1,3,5,7 --> P₁ 0 1 1 --> 0 0 1 1 ----> P₁ = 0

P₂ --> 2,3,6,7 --> P₂ 0 1 1 --> 0 0 1 1 ----> P₂ = 0

P₃ --> 4,5,6,7 --> P₃ 1 1 1 --> 1 1 1 1 ----> P₃ = 1

c).

from the above step we have parity bits P₁ = 0,  P₂ = 0, P₃ = 1

So, the final message bits with parity bits is 0001111

d).

To identify that the received message has error calculate decimal value of P₃P₂ P₁ = (100)₂ = 4

The 4th bit has error

So, to correct the final message invert the 4th bit

So, now the final message becomes 0000111

e).

Here we are asked to find out error position by injecting(Inverting the 3rd position of initial message )

So, now we have to repeat the steps above again.

Step1:

1	2	3	4	5	6	7
P1	P2	m1	P3	m2	m3	m4
P1	P2	1	P3	1	1	1

Now we will calculate the even parity using Exclusive - OR

P₁ --> 1,3,5,7 --> P₁ 1 1 1 --> 1 1 1 1 ----> P₁ = 1

P₂ --> 2,3,6,7 --> P₂ 1 1 1 --> 1 1 1 1 ----> P₂ = 1

P₃ --> 4,5,6,7 --> P₃ 1 1 1 --> 1 1 1 1 ----> P₃ = 1

P₃P₂P₁ = 1 1 1 = 7

Therefore the 7th bit has error