Hamming code

A family of binary linear error-correcting codes that corrects any single-bit error and detects any double-bit error. Its parity checks locate a flipped bit through a syndrome.

Parity-check matrix: A matrix whose multiplication by a received word produces its error syndrome. Its columns assign distinct syndromes to the bit positions in a Hamming code.
Binary field: The two-element field whose arithmetic is addition and multiplication modulo two. Hamming code calculations use this field, where addition is XOR.
Richard Hamming: An American mathematician and computer scientist who developed Hamming codes and advanced numerical analysis. He devised the codes after encountering uncorrectable errors in punched-card work.
ECC memory: Computer memory that uses error-correcting codes to detect or correct data corruption. Many ECC memory systems use an extended Hamming code to correct single-bit errors.
Repetition code: A code that represents each bit by repeating it a fixed number of times. It corrects errors through redundant copies rather than Hamming’s overlapping parity checks.
Syndrome decoding: A decoding method that uses parity-check results to infer an error pattern. A nonzero syndrome identifies which single bit to flip in standard Hamming decoding.
Linear code: A set of codewords forming a vector subspace over a finite field. Hamming codes are linear, so sums of valid codewords remain valid codewords.
Bell Labs: A research and development organization founded in the United States in 1925. Hamming developed his error-correcting codes while working there.
RAID 2: A RAID level that stripes data at the bit level and uses Hamming-code disks for error correction. It is a direct, historically notable application of Hamming coding.
BCH code: A class of cyclic error-correcting codes designed to correct multiple errors. BCH codes generalize the design goal when correcting more than one error is needed.

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