Reed–Solomon error correction
In information theory and coding theory, 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, including consumer technologies such as MiniDiscs, CDs, DVDs, Blu-ray discs, QR codes, Data Matrix, 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 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 check symbols to the data, a Reed–Solomon code can detect any combination of up to erroneous symbols, or locate and correct up to erroneous symbols at unknown locations. As an erasure code, it can correct up to 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 consecutive bit errors can affect at most two symbols of size. The choice of 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.
History
Reed–Solomon codes were developed in 1960 by Irving S. Reed and Gustave Solomon, who were then staff members of MIT Lincoln Laboratory. Their seminal article was titled "Polynomial Codes over Certain Finite Fields". The original encoding scheme described in the Reed and Solomon article used a variable polynomial based on the message to be encoded where only a fixed set of values to be encoded are known to encoder and decoder. The original theoretical decoder generated potential polynomials based on subsets of k out of n values of a received message, choosing the most popular polynomial as the correct one, which was impractical for all but the simplest of cases. This was initially resolved by changing the original scheme to a BCH-code-like scheme based on a fixed polynomial known to both encoder and decoder, but later, practical decoders based on the original scheme were developed, although slower than the BCH schemes. The result of this is that there are two main types of Reed–Solomon codes: ones that use the original encoding scheme and ones that use the BCH encoding scheme.Also in 1960, a practical fixed polynomial decoder for BCH codes developed by Daniel Gorenstein and Neal Zierler was described in an MIT Lincoln Laboratory report by Zierler in January 1960 and later in an article in June 1961. The Gorenstein–Zierler decoder and the related work on BCH codes are described in a book "Error-Correcting Codes" by W. Wesley Peterson. By 1963, J.J. Stone recognized that Reed–Solomon codes could use the BCH scheme of using a fixed generator polynomial, making such codes a special class of BCH codes, but Reed–Solomon codes based on the original encoding scheme are not a class of BCH codes, and depending on the set of evaluation points, they are not even cyclic codes.
In 1969, an improved BCH scheme decoder was developed by Elwyn Berlekamp and James Massey and has since been known as the Berlekamp–Massey decoding algorithm.
In 1975, another improved BCH scheme decoder was developed by Yasuo Sugiyama, based on the extended Euclidean algorithm.
In 1977, Reed–Solomon codes were implemented in the Voyager program in the form of concatenated error correction codes. The first commercial application in mass-produced consumer products appeared in 1982 with the compact disc, where two interleaved Reed–Solomon codes are used. Today, Reed–Solomon codes are widely implemented in digital storage devices and digital communication standards, though they are being slowly replaced by Bose–Chaudhuri–Hocquenghem codes. For example, Reed–Solomon codes are used in the Digital Video Broadcasting standard DVB-S, in conjunction with a convolutional inner code, but BCH codes are used with LDPC in its successor, DVB-S2.
In 1986, an original scheme decoder known as the Berlekamp–Welch algorithm was developed.
In 1996, variations of original scheme decoders called list decoders or soft decoders were developed by Madhu Sudan and others, and work continues on these types of decoders.
In 2002, another original scheme decoder was developed by Shuhong Gao, based on the extended Euclidean algorithm.
Applications
Data storage
Reed–Solomon coding is very widely used in mass storage systems to correctthe burst errors associated with media defects.
Reed–Solomon coding is a key component of the compact disc. It was the first use of strong error correction coding in a mass-produced consumer product, and DAT and DVD use similar schemes. In the CD, two layers of Reed–Solomon coding separated by a 28-way convolutional interleaver yields a scheme called Cross-Interleaved Reed–Solomon Coding. The first element of a CIRC decoder is a relatively weak inner Reed–Solomon code, shortened from a code with 8-bit symbols. This code can correct up to 2 byte errors per 32-byte block. More importantly, it flags as erasures any uncorrectable blocks, i.e., blocks with more than 2 byte errors. The decoded 28-byte blocks, with erasure indications, are then spread by the deinterleaver to different blocks of the outer code. Thanks to the deinterleaving, an erased 28-byte block from the inner code becomes a single erased byte in each of 28 outer code blocks. The outer code easily corrects this, since it can handle up to 4 such erasures per block.
The result is a CIRC that can completely correct error bursts up to 4000 bits, or about 2.5 mm on the disc surface. This code is so strong that most CD playback errors are almost certainly caused by tracking errors that cause the laser to jump track, not by uncorrectable error bursts.
DVDs use a similar scheme, but with much larger blocks, a inner code, and a outer code.
Reed–Solomon error correction is also used in parchive files which are commonly posted accompanying multimedia files on USENET. The distributed online storage service Wuala also used Reed–Solomon when breaking up files.
Bar code
Almost all two-dimensional bar codes such as PDF-417, MaxiCode, Datamatrix, QR Code, Aztec Code and Han Xin code use Reed–Solomon error correction to allow correct reading even if a portion of the bar code is damaged. When the bar code scanner cannot recognize a bar code symbol, it will treat it as an erasure.Reed–Solomon coding is less common in one-dimensional bar codes, but is used by the PostBar symbology.
Data transmission
Specialized forms of Reed–Solomon codes, specifically Cauchy-RS and Vandermonde-RS, can be used to overcome the unreliable nature of data transmission over erasure channels. The encoding process assumes a code of RS which results in N codewords of length N symbols each storing K symbols of data, being generated, that are then sent over an erasure channel.Any combination of K codewords received at the other end is enough to reconstruct all of the N codewords. The code rate is generally set to 1/2 unless the channel's erasure likelihood can be adequately modelled and is seen to be less. In conclusion, N is usually 2K, meaning that at least half of all the codewords sent must be received in order to reconstruct all of the codewords sent.
Reed–Solomon codes are also used in xDSL systems and CCSDS's Space Communications Protocol Specifications as a form of forward error correction.
Space transmission
One significant application of Reed–Solomon coding was to encode the digital pictures sent back by the Voyager program.Voyager introduced Reed–Solomon coding concatenated with convolutional codes, a practice that has since become very widespread in deep space and satellite communications.
Viterbi decoders tend to produce errors in short bursts. Correcting these burst errors is a job best done by short or simplified Reed–Solomon codes.
Modern versions of concatenated Reed–Solomon/Viterbi-decoded convolutional coding were and are used on the Mars Pathfinder, Galileo, Mars Exploration Rover and Cassini missions, where they perform within about 1–1.5 dB of the ultimate limit, the Shannon capacity.
These concatenated codes are now being replaced by more powerful turbo codes:
| Years | Code | Mission |
| 1958–present | Uncoded | Explorer, Mariner, many others |
| 1968–1978 | convolutional codes | Pioneer, Venus |
| 1969–1975 | Reed–Muller code | Mariner, Viking |
| 1977–present | Binary Golay code | Voyager |
| 1977–present | RS + CC | Voyager, Galileo, many others |
| 1989–2003 | RS + CC | Voyager |
| 1989–2003 | RS + CC | Galileo |
| 1996–present | RS + CC | Cassini, Mars Pathfinder, others |
| 2004–present | Turbo codes | Messenger, Stereo, MRO, MSL,others |
| est. 2009 | LDPC codes | Constellation, M2020, MAVEN |