A 4 bit message b = [b_1, b_2, b_3, b_4] is to be sent via a noisy channel. Thre

Question

A 4 bit message b = [b_1, b_2, b_3, b_4] is to be sent via a noisy channel. Three parity bits are employed p_1 = (b_1 + b_2 + b_4) mod 2, p_2 = (b_1 + b_3 + b_4) mod 2 and p_3 = (b_2 + b_3 + b_4) mod 2. The message plus parity bits is sent as a 7 bit message c = [c_1, ..., c_7] where c_1 = p_1, c_2 = p_2, c_3 = b_1, c_4 = p_3, c_5 = b_2, c_6 = b_3 and c_7 = b_4. If the message received at the destination is C_rec = [0, 1, 0, 1, 1, 0, 0], what was the original 4 bit message? Assume that there can be at most one erroneous bit.

Explanation / Answer

So, I am solving it as per my thought process....

Here, you are given the 7-bits in crecwhich means the message plus the parity bits received at receiver.

Here, the equations are:

b = b1 b2 b3 b4

p1 = (b1 + b2 + b4) mod 2

p2 = (b1 + b3 + b4) mod 2

p3 = (b2 + b3 + b4) mod 2

The data given for c is :

c1 = p1 = 0

c2 = p2 = 1

c3 = b1 = 0

c4 = p3 = 1

c5 = b2 = 1

c6 = b3 = 0

c7 = b4 = 0

Now, let's calculate the p1, p2, and p3 from the given b1, b2, b3 and b4 values.

p1 = (b1 + b2 + b4) mod 2 = ( 0 + 1 + 0 ) mod 2 = 1 mod 2 = 1

p2 = (b1 + b3 + b4) mod 2 = ( 0 + 0 + 0 ) mod 2 = 0 mod 2 = 0

p3 = (b2 + b3 + b4) mod 2 = ( 1 + 0 + 0 ) mod 2 = 1 mod 2 = 1

So, from b1 to b4, the p1, p2 and p3 values are 1, 0 , and 1. But, the received values of p1, p2 and p3 are : 0, 1 and 1.

So, at the receiver side, the p1 and p2 value is changed that means the content which is common in them must be erroneous. Now, both are having b1 and b4 in common. So, error must be in either b1 or b4 because that are common in the both p1 and p2 so the value change in p is either because of b1 or b4.

Now, as p3 is not having any error that means b2, b3 and b4 are correct because it is mentioned that at max 1 bit is erroneous. So, from b1 and b4 also, only b1 is having error.

So, receiver receives 0 as b1 but it is having error so the original value is 1.

So, the original 4-bit number is 1100.

Check the equations now and then p1, p2 and p3 all are having the correct values. So, b1 was erroneous and the original message is 1100.

Please comment if there is any query. Thank you. :)

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