Question

What should be the minimum Hamming distance \( d_{\text{min}} \) to guarantee correction of up to p errors in a given block code?

• A. \( 2p \)
• B. \( 2p + 1 \)
• C. \( 2p - 1 \)
• D. \( 2^{p} \)

Hint:

For a code to correct \( p \) errors, the minimum Hamming distance must be at least \( 2p + 1 \).

Answer 1

The Correct Option is B

Step 1: Defining Hamming Distance.
The Hamming distance quantifies the number of differing positions between corresponding symbols of two code words. To ensure the correction of up to p errors, the minimum Hamming distance \( d_{\text{min}} \) must adhere to the condition: \[d_{\text{min}} \geq 2p + 1\]This inequality is a prerequisite for a code to be able to correct p errors.

Step 2: Final Determination.
Consequently, the accurate selection is (2) \( 2p + 1 \).