Reed–Solomon error correction
Reed–Solomon codes are a group of error-correcting codes that were introduced by Irving S. Reed and Gustave Solomon in 1960. They have many applications, the most prominent of which include consumer technologies such as MiniDiscs, CDs, DVDs, Blu-ray discs, QR codes, data transmission technologies such as DSL and WiMAX, broadcast systems such as satellite communications, DVB and ATSC, and storage systems such as RAID 6.
| Reed–Solomon codes | |
|---|---|
| Named after | Irving S. Reed and Gustave Solomon |
| Classification | |
| Hierarchy | Linear block code Polynomial code Reed–Solomon code |
| Block length | n |
| Message length | k |
| Distance | n − k + 1 |
| Alphabet size | q = pm ≥ n (p prime) Often n = q − 1. |
| Notation | [n, k, n − k + 1]q-code |
| Algorithms | |
| Berlekamp–Massey Euclidean et al. | |
| Properties | |
| Maximum-distance separable code | |
Reed–Solomon codes operate on a block of data treated as a set of finite-field elements called symbols. Reed–Solomon codes are able to detect and correct multiple symbol errors. By adding t = n − k check symbols to the data, a Reed–Solomon code can detect (but not correct) any combination of up to t erroneous symbols, or locate and correct up to ⌊t/2⌋ erroneous symbols at unknown locations. As an erasure code, it can correct up to t erasures at locations that are known and provided to the algorithm, or it can detect and correct combinations of errors and erasures. Reed–Solomon codes are also suitable as multiple-burst bit-error correcting codes, since a sequence of b + 1 consecutive bit errors can affect at most two symbols of size b. The choice of t is up to the designer of the code and may be selected within wide limits.
There are two basic types of Reed–Solomon codes – original view and BCH view – with BCH view being the most common, as BCH view decoders are faster and require less working storage than original view decoders.