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.

What is the Hamming Distance Calculator?

Calculate the Hamming distance between two integers — the number of bit positions where they differ. Enter values in binary, decimal, or hex and see the distance, the aligned binary forms, and the XOR. Useful for error-correcting codes and hashing. Runs in your browser. It runs free in your browser on Gera Tools, with nothing uploaded.

Hamming Distance Calculator

Name: Hamming Distance Calculator
Creator: Gera Tools
License: https://creativecommons.org/licenses/by/4.0/

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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.

Worked example

Comparing the binary values 10110110 and 11100100:

  10110110
  11100100
  --------
XOR 01010010

The XOR result 01010010 has three 1 bits at positions 1, 4, and 6 (counting from the right). The Hamming distance is therefore 3.

In decimal, that is comparing 182 and 228 — their XOR is 82, and the popcount of 82 (binary 01010010) is 3.

Practical uses

Error-correcting codes: The fundamental idea in codes like Hamming(7,4) and BCH codes is to ensure that any two valid codewords differ in at least a minimum number of bit positions. A minimum distance of 3 allows single-bit error detection and correction: if you receive a word, the nearest valid codeword (the one with the smallest Hamming distance) is the most likely intended value. Distance 4 allows detecting two-bit errors.

Perceptual image hashing: A perceptual hash (pHash, dHash) turns an image into a 64-bit integer that changes smoothly as the image changes. Two hashes with a Hamming distance of 0 are identical images; a distance below about 10 typically indicates the same image with minor edits, resizing, or compression artefacts. Distance above 20–30 usually means different images.

DNA and protein sequences: Mutations in a fixed-length sequence are exactly bit-position changes if you encode the bases as bits. Hamming distance counts the number of substitution mutations between two sequences of the same length (but not insertions or deletions, which require edit distance instead).

Network IP addressing: Comparing subnet masks or checking prefix lengths sometimes involves counting differing bits, though this is more commonly framed as bitwise AND operations.

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.