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31 votes
31 votes
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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Best answer
27 votes
27 votes

Here, answer is yes. There is error in This message. Error is in bit $6$.

How to calculate it? First of all reverse given input to get it in correct position from $1$ to $7$.

$0110001$

$\text{Bit-1, Bit-2 & Bit-4}$ are partity bits.

Calculating position of error $\Rightarrow$

$\text{$\large c_4 c_2 c_1$}$

$\text{1   1   0}$

Here, $c4 = \text{bit}4\oplus \text{bit}5 \oplus \text{bit}6 \oplus \text{bit}7=0\oplus 0\oplus 0\oplus 1=1$
                   $\text{ (Taking Even parity )}$

          $c2  = \text{bit}2\oplus \text{bit}3\oplus \text{bit}6\oplus \text{bit}7=1\oplus 1\oplus 0\oplus 1=1$

          $c1= \text{bit}1\oplus \text{bit}3\oplus \text{bit}5\oplus \text{bit}7=0\oplus 1\oplus 0\oplus 1=0$

Reference: $\Rightarrow$ https://en.wikipedia.org/wiki/Hamming%287,4%29

When you correct bit $6$.

You get a message as $0110011$.

If you calculate $\large c_4,c_2,c_1$ all will be $0$ now!

edited by
8 votes
8 votes

Yes there is an error in this message at 6-bit position so the correct message is 1100110

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