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: Understanding Hamming Distance.
The **Hamming distance** is the number of positions at which the corresponding symbols are different between two code words. To guarantee correction of **up to p errors**, the minimum Hamming distance \( d_{\text{min}} \) must satisfy: \[ d_{\text{min}} \geq 2p + 1 \] This ensures that the code is capable of correcting **p errors**.

Step 2: Conclusion.
Thus, the correct answer is **(2) \( 2p + 1 \)**.