Hamming code: Difference between revisions

Revision as of 05:11, 12 November 2024

Revision as of 12:32, 8 December 2024

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Shown are only 20 encoded bits (5 parity, 15 data) but the pattern continues indefinitely. The key thing about Hamming Codescodes that can be seen from visual inspection is that any given bit is included in a unique set of parity bits. To check for errors, check all of the parity bits. The pattern of errors, called the [[Syndrome decoding|error syndrome]], identifies the bit in error. If all parity bits are correct, there is no error. Otherwise, the sum of the positions of the erroneous parity bits identifies the erroneous bit. For example, if the parity bits in positions 1, 2 and 8 indicate an error, then bit 1+2+8=11 is in error. If only one parity bit indicates an error, the parity bit itself is in error.

With {{mvar|m}} parity bits, bits from 1 up to <math>2^m-1</math> can be covered. After discounting the parity bits, <math>2^m-m-1</math> bits remain for use as data. As {{mvar|m}} varies, we get all the possible Hamming codes: