Question

Given the following Hamming-encoded codeword received at receiver as "01100001111" having 7 bits of data, 4 bits of error correction code. Determine if the data has been received correctly, and if it is incorrect (assuming that no more than a single bit has changed):

Answer 1

Given,

Hamming code = 0 1 1 0 0 0 0 1 1 1 1

Data bit = 7

Parity bit = 4

$2^0=1 = P_1,\\ 2^1=2=P_2, \\2^2= 4=P_4,\\ 2^3=8=P_8$

Rest of all are data bit,

P1: 1, 3, 5, 7, 9, 11=110010 (odd)

So, it is contradiction, hence $P_1=1$

P2: 2, 3, 6, 7, 10, 11 = 110010 (odd)

So, it is contradiction, hence $P_2=1$

 P4: 4, 5, 6, 7=1000

it is also a contradiction so $P_4=1$

$P_8: 8, 9, 10, 11=0110$

even, so no error is detected. Hence, $P_8=0$

Now, writing it collectively, $(P_8P_4P_2P_1)$

$(0111)_2 =7$

Here, 7th bit is corrupted. So correct code is 0 1 1 0 1 0 0 1 1 1 1