ReedSolomonkódok

Reed-Solomon kódok, or Reed-Solomon codes, are a family of error-correcting codes that operate on symbols from a finite field to detect and correct errors in blocks of data. They are named after Irving Reed and Gustave Solomon, who introduced them in 1960. RS codes are maximum distance separable (MDS) codes, meaning they achieve the largest possible minimum distance for a given block length. An RS code is denoted RS(n, k), where n is the block length and k is the number of data symbols; the code adds n−k parity symbols, enabling correction of up to t = (n−k)/2 symbol errors within a block. The symbols come from a finite field, typically GF(2^m), with n ≤ 2^m − 1.

Encoding is performed by multiplying the data symbol vector by a generator polynomial of degree n−k, or equivalently by evaluating a message polynomial at a set of distinct field elements. Decoding begins with calculating syndromes to detect errors and then applying algorithms such as the Berlekamp–Madde algorithm and the Sugiyama approach to locate error positions and determine their magnitudes, with Forney’s algorithm commonly used to compute the actual error values.

RS codes are widely used in storage and communications to protect against burst errors. They are integral to the error-correction schemes of compact discs, DVDs, and Blu-ray discs, and they underpin the error resilience of QR codes, data transmissions, and various archival storage systems.

See also: error-correcting codes, finite fields, maximum distance separable codes.