Error-correcting code

An encoding method that adds structured redundancy to data so errors introduced during storage or transmission can be detected or corrected.

Hamming distance: The number of positions at which two equal-length strings differ. The minimum distance between codewords determines how many errors a code can detect or correct.
Finite field: A field with finitely many elements, supporting addition, subtraction, multiplication, and division by nonzero elements. Many practical codes perform their arithmetic over finite fields.
Reed–Solomon code: A nonbinary error-correcting code that represents messages as evaluations of polynomials over a finite field. It corrects symbol errors and underlies many storage and communication systems.
Automatic repeat request: A communication protocol that detects errors and requests retransmission of damaged or missing data. It trades retransmission for correction, often combining with codes in hybrid systems.
Parity bit: A bit added to data to make the number of 1 bits even or odd. A simple parity check shows how redundancy can detect errors without correcting them.
Linear code: A code whose codewords form a vector subspace over a finite field. Linearity enables compact matrix descriptions and efficient encoding and decoding.
Low-density parity-check code: A linear code defined by a sparse parity-check matrix and commonly decoded iteratively. Its strong performance near channel capacity makes it useful in modern communications.
Forward error correction: Error correction performed by adding redundancy before transmission, without relying on retransmission. It is the no-retransmission use of error-correcting codes when feedback is unavailable or costly.
Syndrome decoding: A decoding method that uses parity-check results to identify an error pattern. A syndrome summarizes violated checks and guides correction in many linear codes.
Information theory: The mathematical study of information, communication, compression, and uncertainty. Coding theory uses its limits on reliable communication through noisy channels.

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