Hamming Code Calculator

Hamming codes add parity bits so a one-bit error can be detected and corrected.

Use the result as a learning aid. For classwork, still show the formula and intermediate reasoning so the final answer is not just a black-box number.

Hamming(7,4) turns four data bits into a seven-bit codeword by adding three parity bits. Using one-based positions, parity bits occupy positions 1, 2, and 4, while data bits occupy positions 3, 5, 6, and 7. Each parity bit checks a different overlapping set of positions.

With even parity, p1 checks positions 1,3,5,7; p2 checks 2,3,6,7; and p4 checks 4,5,6,7. At the receiver, the same parity checks form a three-bit syndrome. Interpreted as a binary index, a nonzero syndrome points to the position of a possible single-bit error.

Standard Hamming(7,4) has minimum distance 3, so it can correct one bit error in a codeword. It does not reliably correct arbitrary two-bit errors. Extended Hamming codes add an overall parity bit to improve double-error detection.

Once a seven-bit word is formed, verify each even-parity group independently. Count the 1s in positions 1,3,5,7; then 2,3,6,7; then 4,5,6,7. Every count should be even for a valid even-parity Hamming(7,4) codeword. This gives a transparent manual check of the three parity equations.

For a received word, the three failed/pass results form the syndrome. The p1 check is the 1s place, p2 the 2s place, and p4 the 4s place. For example, failed p1 and p4 but passed p2 gives syndrome 1+4=5, pointing to position 5 under the single-error assumption. After flipping the indicated bit, recompute all three checks; a corrected single-bit error should leave syndrome 0.