What will be the minimum Hamming distance for the following coding scheme?

Concept: The minimum Hamming distance (dmin) of a coding scheme is the smallest number of bit positions in which any two distinct valid codewords differ. The Hamming distance between two equal-length codewords is the count of positions where their bits disagree.

Application: The four valid codewords are 00111, 01001, 10101 and 11100. Compare every one of the six possible pairs and count the differing positions:

1. 00111 vs 01001: differ at positions 2, 3, 4 → distance = 3

2. 00111 vs 10101: differ at positions 1, 4 → distance = 2

3. 00111 vs 11100: differ at positions 1, 2, 4, 5 → distance = 4

4. 01001 vs 10101: differ at positions 1, 2, 3 → distance = 3

5. 01001 vs 11100: differ at positions 1, 3, 5 → distance = 3

6. 10101 vs 11100: differ at positions 2, 5 → distance = 2

Cross-check: The smallest of the six distances {3, 2, 4, 3, 3, 2} is 2, achieved by the pairs 00111–10101 and 10101–11100. No pair differs in only one position, so the distance cannot be 1.

Result: the minimum Hamming distance of this coding scheme is 2.