GATE CS 2021 Set 1 — Question 39
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Computer Networks → Data Link Layer → Error Correction (Hamming)
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Question
Assume that a 12-bit Hamming codeword consisting of 8-bit data and 4 check bits is , where the data bits and the check bits are given in the following tables:Data bits
Check bits
Which one of the following choices gives the correct values of and ?
| 1 | 1 | 0 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 |
Correct answer
(A) x is 0 and y is 0.
Solution
In a standard Hamming code, the check bits are located at positions that are powers of 2 (). The given sequence is . Mapping these to positions 12 down to 1 (left to right):
Bits:
Values:
Sum for even parity: .Check bit (Position 2): Checks positions with the 2nd bit set (2, 3, 6, 7, 10, 11).
Bits:
Values:
Sum: . (Consistent)Check bit (Position 4): Checks positions with the 3rd bit set (4, 5, 6, 7, 12).
Bits:
Values:
Sum: . (Consistent)Check bit (Position 8): Checks positions with the 4th bit set (8, 9, 10, 11, 12).
Bits:
Values:
Substituting :
Sum for even parity: .Thus, and .
- Position 12:
- Position 11:
- Position 10:
- Position 9:
- Position 8:
- Position 7:
- Position 6:
- Position 5:
- Position 4:
- Position 3:
- Position 2:
- Position 1:
Bits:
Values:
Sum for even parity: .Check bit (Position 2): Checks positions with the 2nd bit set (2, 3, 6, 7, 10, 11).
Bits:
Values:
Sum: . (Consistent)Check bit (Position 4): Checks positions with the 3rd bit set (4, 5, 6, 7, 12).
Bits:
Values:
Sum: . (Consistent)Check bit (Position 8): Checks positions with the 4th bit set (8, 9, 10, 11, 12).
Bits:
Values:
Substituting :
Sum for even parity: .Thus, and .
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