ReedSolomondekodning

Reed-Solomon decoding refers to the set of methods used to recover original data from symbols encoded with Reed-Solomon codes, a family of non-binary error-correcting codes defined over finite fields. These codes treat data as symbols taken from a finite field, allowing efficient correction of burst and random symbol errors.

A Reed-Solomon code is defined by parameters [n, k] over GF(q), with n ≤ q − 1 and minimum distance d = n − k + 1. Consequently, it can correct up to t = floor((d − 1)/2) symbol errors in a codeword of length n. In practice, many RS codes operate over GF(2^8), making each symbol a byte, which facilitates hardware and software implementations.

The most common decoding approach is syndrome-based. After receiving a word, the decoder computes syndromes from the received symbols. It then determines the error locator polynomial, typically using the Berlekamp-Massey algorithm or an equivalent method. A Chien search locates the error positions by finding the roots of this polynomial, and Forney’s algorithm computes the corresponding error magnitudes to correct the symbols. The result is a corrected codeword, from which the original data can be retrieved.

Extensions and variants include list decoding, such as the Guruswami-Sudan algorithm, which can decode beyond half the minimum distance and produce a small list of candidate messages. Soft-decision and iterative decoding methods also improve performance in noisy channels.

Applications of Reed-Solomon decoding are widespread in data storage and transmission, including CDs, DVDs, Blu-ray discs, QR codes, RAID systems, and various forms of digital communication where burst errors are common.