Question

4. Using Hamming code, assume the sender want to send a message of 4 bits, 1111.

(a) What is actually being transmitted

(b) Assume during transmission the parity bit p1 is inverted by an error (noise)

i. How does the receiver detect the error? ii. How does the receiver correct the error?

Show your work by calculating parity bits and syndrome bits.

Answer 1

a) With 4 data bits we need 3 parity bits.

p1 = d1 + d2 + d4
p2 = d1 + d4 + d3
p3 = d2 + d4 + d3

+ is XOR operation here.

The transmitted string will be d₁d₂d₃d₄p₁p₂p₃.

So the three parity bits will be

p1 = 1 + 1 + 1 = 1
p2 = 1 + 1 + 1 = 1
p3 = 1 + 1 + 1 = 1

So transmitted data will be all 1's, i.e., 1111111.

b) i)It is said that p₁ gets flipped so the received data will be 1111011.

Now here again we check parities:-

p1 = 1 + 1 + 1 = 1 (but p1 is 0 here)
p2 = 1 + 1 + 1 = 1
p3 = 1 + 1 + 1 = 1

Error detected successfully.

ii) If it would have been an error in one of the data bits then it would have been reflected in two or more parity checks but there is an error in only one calculation, which means the parity bit p1 was flipped. So to correct it we can simply flip the parity bit again. It really doesn't matter because we have correctly received all the data bits.

Hope this helps. If you have any queries or suggestions regarding the answers please leave them in the comments section so I can update and improve the answer. Thank you