Hamming code calculator

Type data bits in groups of four. Each group becomes seven bits, with the three parity bits worked out in the steps. Swap the boxes to check a code: flip any one bit of a seven-bit group and the decoder finds it, fixes it, and gives the original four data bits back.

A parity bit is one extra bit that makes the count of 1s even. Send 1011 with even parity and it goes as 1011 1; if any single bit flips on the way, the count turns odd and the receiver knows something went wrong. It can't tell which bit, though, and two flips cancel out.

Number the seven positions 1 to 7. Parity bits sit at the powers of two, 1, 2 and 4; data bits fill 3, 5, 6 and 7. Each parity bit checks the positions whose number includes it in binary: p1 checks 1, 3, 5, 7; p2 checks 2, 3, 6, 7; p4 checks 4, 5, 6, 7.

When a bit flips, the groups that now fail add up to its position. If p1 and p4 fail, the bad bit is 1 + 4 = 5. Flip it back and the data is right again. Richard Hamming worked this out at Bell Labs and published it in 1950, after his weekend jobs on a relay computer kept being dropped: the machine could detect an error, but with no operator there it simply moved on to the next job.