Binary Hamming Distance Form
Example Data Table
| Binary A | Binary B | Mode | Distance | XOR Pattern | Meaning |
|---|---|---|---|---|---|
| 10110101 | 10011101 | Equal length | 2 | 00101000 | Two bit positions differ. |
| 1110 | 1010 | Equal length | 1 | 0100 | One received bit may be flipped. |
| 101 | 000101 | Left pad | 0 | 000000 | Values match after leading zeros. |
| 11001 | 10011 | Equal length | 3 | 01010 | Patterns show moderate disagreement. |
Formula Used
For two aligned binary strings A and B, the Hamming distance is: d(A,B) = Σ [Aᵢ ≠ Bᵢ]. Each different bit adds one. Each matching bit adds zero.
Normalized distance is d / n, where n is the compared bit length. Similarity is 1 - d / n. The XOR pattern is useful because every one in XOR marks a mismatch.
The calculator also shows hamming weight. That is the count of one bits in a string. If a distance is treated like a minimum code distance proxy, detection is d - 1. Correction is floor((d - 1) / 2). A single pair comparison is still not a full codebook proof.
How to Use This Calculator
- Enter the first binary string in the first box.
- Enter the second binary string in the second box.
- Choose equal length, left padding, or right padding.
- Select position numbering and direction.
- Set a bit group size for easier reading.
- Press Calculate to view the result below the header.
- Use CSV or PDF export when records are needed.
Binary Distance in Physics Workflows
Hamming distance counts how many positions differ between two binary words. In physics, those words can represent sampled states, detector flags, encoded telemetry, or digital signals. A small distance means the patterns are close. A larger distance shows stronger disagreement, noise, or possible transmission error.
Why the Measurement Matters
Binary measurements often travel through sensors, cables, satellites, or storage systems. Each stage can flip a bit. The calculator helps compare an expected pattern with a received pattern. It shows the exact mismatch positions, so a lab user can trace where the changes happened. It also gives a normalized distance, which works like a bit error rate for one comparison.
Advanced Comparison Options
The tool supports equal length comparison and zero padding. Equal length mode is best when both words are fixed codewords. Left padding is useful when numbers are written without leading zeros. Right padding can help with stream fragments or aligned detector windows. The XOR pattern gives a direct visual map. A one marks a different bit. A zero marks a matching bit.
Formula and Interpretation
The main formula adds one for every position where the two bits are different. Matching positions add zero. Dividing that value by the compared length gives the normalized distance. Subtracting the normalized distance from one gives similarity. The hamming weight of each word counts the number of ones. These values help describe signal density and parity behavior.
Practical Use
Enter two binary strings, choose the alignment mode, and submit the form. Spaces may be ignored, so grouped bits are easy to paste. Review the result panel first. It appears directly below the header for quick reading. Download the CSV for spreadsheets. Download the PDF for reports, lab notes, or teaching material. Use the example table to check expected behavior before comparing real data. In coding studies, remember that one pair distance is not a full code minimum. It still gives a useful local error measure.
Quality Checks
For best results, keep notes about the source of each string. Record sampling rate, channel name, and expected code length. Repeat comparisons after filtering or decoding. Consistent distance changes can reveal drift, timing faults, or noisy hardware before larger failures appear.
FAQs
What is Hamming distance?
Hamming distance is the number of positions where two equal length strings differ. For binary data, it counts how many bits changed between two compared words.
Can the calculator compare unequal binary strings?
Yes. Use left padding or right padding when the strings have different lengths. Equal length mode keeps strict comparison and reports an error if lengths differ.
What does the XOR pattern show?
The XOR pattern marks mismatches. A one means the two bits are different. A zero means the two bits are the same at that position.
Why is normalized distance useful?
Normalized distance divides the mismatch count by total compared bits. It helps compare short and long strings on the same scale.
What is similarity percentage?
Similarity percentage is one minus normalized distance, then multiplied by 100. Higher values mean the two binary strings are more alike.
What is hamming weight?
Hamming weight is the number of one bits in a binary string. It helps describe density, parity, and signal activity.
Can this tool detect transmission errors?
It can show differences between expected and received binary data. Actual error detection strength depends on the full coding scheme, not one pair alone.
Why use padding?
Padding aligns strings before comparison. Left padding is common for binary numbers. Right padding can help compare stream fragments or fixed sampling windows.