What should be the minimum hamming distance \( d_{min} \) to guarantee correction of upto p errors in a given block code.
Step-by-step Solution:
The minimum Hamming distance of a block code, denoted as d_{min}, determines its error-detecting and error-correcting capability.
- To detect up to p errors, the minimum distance must satisfy: d_{min} \geq p + 1.
- To correct up to p errors, the minimum distance must satisfy: d_{min} \geq 2p + 1.
This is because with distance 2p + 1, the codewords are far enough apart that even if p errors occur, the received word is still closer to the correct codeword than to any other.
✅ Correct Answer: (B) 2p + 1