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Following $7$ bit single error correcting hamming coded message is received.
$$\overset{7\qquad 6\qquad 5 \qquad 4\qquad 3 \qquad 2 \qquad 1}{\boxed{1 \qquad 0\qquad 0 \qquad 0 \qquad 1  \qquad 1 \qquad 0}} \qquad \overset{\textbf{bit No.}}{\boxed{X}}$$
Determine if the message is correct (assuming that at most $1$ bit could be corrupted). If the message contains an error find the bit which is erroneous and gives correct message.
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   d7   d6  d5   p4   d3  p2   p1
   1    0   0    0   1   1   0

 

from the above table,

p1=0

p2=1

p4=0

now use parity bits to check whether there is any error

p1=d3 d5 d7 =1 0 1 =  0  (correct),

  means d3 d5 d7 are correct

p2= d3 d6 d7 = 1 0 1 = 0  (wrong), which is actually 1

 means any of d3 d6 d7 is wrong, but we know from p1 that d3 & d7 are correct. i,e d6 is wrong

so, error found at 6th position

 

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Number the bits as given in the question :

Now take only those number in which bit is 1 ( in this case 7, 3, 2). Write them in binary and apply XOR. If we get 0 there is no error. If we get a number that is the bit where error is present.

7 – 111

3 – 011

2 – 010

Result : 110( which is 6).

So error is present in bit 6.
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–1 votes
Answer: Bit 5 has the error. It should be 1.

Here, parity bits are 1,2,4 (power of 2).

Hamming code based on data bits received = 110
Bit 1 = Bit 3 $\oplus$ Bit 5 = 1 $\oplus$ 0 = 1. (XOR of all bits having the least significant digit as 1)
Bit 2 = Bit 3 $\oplus$ Bit 6 = 1 $\oplus$ 0 = 1. (XOR of all bits having the second least significant digit as 1)
Bit 7 = Bit 4 $\oplus$ Bit 5 $\oplus$ Bit 6 = 0 $\oplus$ 0 $\oplus$ 0 = 0. (XOR of all bits having the third least significant digit as 1)

But the received message has Bit 1 = 0, Bit 2 = 1, Bit 7 = 1 (Actual hamming code)

Bit error = Hamming code based on data bits received $\oplus$ Actual hamming code = 110 $\oplus$ 011 = 101 (i.e. Bit 5 has the error)

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