1. What is Minimum Hamming Distance?

The minimum Hamming distance of a code is the smallest number of bit positions in which any two distinct codewords differ. It's a crucial parameter in coding theory that determines the error detection and correction capabilities of a code.

2. How Does the Calculator Work?

The calculator uses the Hamming distance formula:

Where:

• \( d_H \) — Hamming distance between two codewords
• \( c_i, c_j \) — Codewords from the code C
• \( d_{min} \) — Minimum Hamming distance of the code

Explanation: The calculator computes the Hamming distance between all pairs of codewords and returns the smallest value found.

3. Importance of Hamming Distance

Details: The minimum Hamming distance determines:

• Error detection capability: Can detect up to \( d_{min}-1 \) errors
• Error correction capability: Can correct up to \( \lfloor(d_{min}-1)/2\rfloor \) errors
• Code performance: Larger distances provide better error resilience

4. Using the Calculator

Tips:

• Enter binary codewords separated by commas (e.g., "1010, 1100, 0110")
• All codewords must be binary strings (contain only 0s and 1s)
• At least two codewords are required

5. Frequently Asked Questions (FAQ)

Q1: What is Hamming distance?
A: Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different.

Q2: Why is minimum Hamming distance important?
A: It determines how many errors a code can detect or correct. Higher minimum distance means better error handling.

Q3: Can I use non-binary codewords?
A: This calculator only works with binary codewords (strings of 0s and 1s).

Q4: What if codewords have different lengths?
A: The calculator automatically pads shorter codewords with leading zeros to match the longest codeword's length.

Q5: What's the maximum number of codewords I can enter?
A: There's no hard limit, but very large numbers may slow down the calculation.