Hamming Code Calculator (Encode, Decode, Correct)

A Hamming code adds a few parity bits to a block of data so that any single flipped bit can be found and corrected. It is the textbook example of error-correcting codes and is still used in ECC memory (as the SECDED variant). This calculator encodes data bits into a codeword, or checks a received word: it computes the syndrome, tells you which bit is wrong, corrects it and extracts the data — with a table showing which positions each parity bit covers.

How to use this tool

1. Choose Encode or Check.
2. Type the data bits (to encode) or the received word (to check).
3. Pick the code size — or let the tool fit it to the length — and even or odd parity.
4. Add the overall parity bit for SECDED if you need to detect double errors.

Formula

parity bit at position 2ᵗ = XOR of all positions with bit t set · syndrome = XOR of the positions of all 1 bits = position of the error

With r parity bits a codeword has n = 2ʳ − 1 positions and carries n − r data bits — (7,4), (15,11), (31,26); for other lengths the smallest r with 2ʳ ≥ k + r + 1 is used. Positions are numbered from 1 on the left and the data bits fill the positions that are not powers of two, in order. With odd parity every check is inverted. SECDED adds one bit with the parity of the whole word: a non-zero syndrome with failing overall parity is a correctable single error; a non-zero syndrome with overall parity intact means two errors.

Example

Encoding 1011 with Hamming(7,4): the data goes to positions 3, 5, 6 and 7; P1 = 1⊕0⊕1 = 0, P2 = 1⊕1⊕1 = 1 and P4 = 0⊕1⊕1 = 0, giving 0110011. If bit 5 flips (0110111), the 1 bits sit at positions 2, 3, 5, 6, 7 and 2⊕3⊕5⊕6⊕7 = 5, so bit 5 is corrected and the data 1011 is recovered.