What is the Hamming distance?

The Hamming distance between two equal-length strings is the number of positions at which the corresponding symbols differ. For binary values it counts how many bit positions are not the same.

How is it calculated from two numbers?

XOR the two values, then take the population count of the result. XOR produces a 1 in exactly the positions where the inputs differ, so counting those 1 bits gives the distance directly.

What is Hamming distance used for?

It defines the error-detecting and error-correcting power of a code, measures similarity between perceptual hashes and fingerprints, and quantifies mutations between DNA or protein sequences encoded as strings.

Do the two numbers need the same number of bits?

Conceptually yes, so the shorter value is treated as having leading zeros. The tool left-pads both binary forms to the same width before comparing, which is the standard convention.

How does distance relate to error correction?

A code with minimum Hamming distance d can detect up to d minus 1 bit errors and correct up to the floor of (d minus 1) divided by 2. Larger minimum distance means more robust transmission.

Hamming Distance Calculator

The Hamming distance counts how many bit positions differ between two values. It is the foundation of error-correcting codes, perceptual-hash comparison, and sequence analysis. This tool computes it for any two non-negative integers, entered in binary, decimal, or hex, and shows exactly where the bits disagree.

How it works

Computing the Hamming distance reduces to two well-known bit operations:

diff     = a XOR b        // 1 wherever the bits differ
distance = popcount(diff) // count those 1 bits

XOR yields a 1 in every position where the two inputs disagree and a 0 where they match, so the population count of the XOR is precisely the number of differing positions. The tool left-pads both numbers to the same bit width before displaying them, matching the convention that the shorter value carries implicit leading zeros.

Example and notes

Comparing 10110110 and 11100100, the XOR is 01010010, which has three 1 bits — so the Hamming distance is 3. In coding theory the minimum Hamming distance of a code sets its strength: a distance of d lets you detect up to d − 1 errors and correct up to (d − 1) / 2. The same measure compares image hashes (a small distance means visually similar) and counts mutations between biological sequences. This calculator uses BigInt, so the values can be arbitrarily large.