A communication system is designed using the following three 5 bit words as…
2025
A communication system is designed using the following three 5 bit words as the only valid codewords and uses error correction using hamming distance.
Codeword A: 00000
Codeword B: 01110
Codeword C: 11000
During operation, receiver gets the word 01000. What would the system conclude?
- A.
It will correct to codeword A
- B.
It will correct to codeword B
- C.
It will correct to codeword C
- D.
It will be unable to correct the error
Attempted by 208 students.
Show answer & explanation
Correct answer: D
Concept
In minimum-distance (nearest-neighbour) decoding, a received word is corrected to the valid codeword that lies at the smallest Hamming distance from it. The Hamming distance between two equal-length bit strings is the number of positions in which they differ. A correction is committed only when exactly one valid codeword is strictly closest; if two or more valid codewords share that smallest distance, no unique nearest codeword exists and the error cannot be resolved unambiguously.
Application
Compute the Hamming distance from the received word 01000 to each valid codeword by counting the differing bit positions:
Codeword | Bits | Differing positions vs 01000 | Hamming distance |
|---|---|---|---|
A | 00000 | position 2 | 1 |
B | 01110 | positions 3 and 4 | 2 |
C | 11000 | position 1 | 1 |
The smallest Hamming distance is 1, and it is achieved by two different valid codewords, 00000 and 11000. Because the minimum distance is shared rather than held by a single codeword, there is no unique nearest codeword to decode to.
Cross-check
For the decoder to commit a correction, the winner must beat every other codeword strictly. Here the closest distance (1) ties between two codewords while the third (01110) sits farther away at distance 2. A tie at the minimum violates the strict-uniqueness requirement, so the system reports that it is unable to correct the error.