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 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.
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". (Reed & Solomon 1960). The original encoding scheme described in the Reed & Solomon article used a variable polynomial based on the message to be encoded where only a fixed set of values (evaluation points) to be encoded are known to encoder and decoder. The original theoretical decoder generated potential polynomials based on subsets of k (unencoded message length) out of n (encoded message length) 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 a paper 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 (1961). By 1963 (or possibly earlier), J. J. Stone (and others) 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 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 more modern low-density parity-check (LDPC) codes or turbo codes. For example, Reed–Solomon codes are used in the Digital Video Broadcasting (DVB) standard DVB-S, but LDPC codes are used 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 – see Guruswami–Sudan list decoding algorithm.
In 2002, another original scheme decoder was developed by Shuhong Gao, based on the extended Euclidean algorithm .
Reed–Solomon coding is very widely used in mass storage systems to correct the 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 (CIRC). The first element of a CIRC decoder is a relatively weak inner (32,28) Reed–Solomon code, shortened from a (255,251) 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 (28,24) 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 (208,192) inner code, and a (182,172) 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 (discontinued in 2015) also used Reed–Solomon when breaking up files.
Almost all two-dimensional bar codes such as PDF-417, MaxiCode, Datamatrix, QR Code, and Aztec 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.
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(N, K) 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.
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 (e.g., direct digital broadcasting) 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:
The Reed–Solomon code is actually a family of codes, where every code is characterised by three parameters: an alphabet size q, a block length n, and a message length k, with k < n ≤ q. The set of alphabet symbols is interpreted as the finite field of order q, and thus, q has to be a prime power. In the most useful parameterizations of the Reed–Solomon code, the block length is usually some constant multiple of the message length, that is, the rate R = k/n is some constant, and furthermore, the block length is equal to or one less than the alphabet size, that is, n = q or n = q − 1.
A discrete Fourier transform is essentially the same as the encoding procedure; it uses the generator polynomial p(x) to map a set of evaluation points into the message values as shown above:
The inverse Fourier transform could be used to convert an error free set of n < q message values back into the encoding polynomial of k coefficients, with the constraint that in order for this to work, the set of evaluation points used to encode the message must be a set of increasing powers of α:
However, Lagrange interpolation performs the same conversion without the constraint on the set of evaluation points or the requirement of an error free set of message values and is used for systematic encoding, and in one of the steps of the Gao decoder.
The Reed–Solomon code properties discussed above make them especially well-suited to applications where errors occur in bursts. This is because it does not matter to the code how many bits in a symbol are in error — if multiple bits in a symbol are corrupted it only counts as a single error. Conversely, if a data stream is not characterized by error bursts or drop-outs but by random single bit errors, a Reed–Solomon code is usually a poor choice compared to a binary code.
The Reed–Solomon code, like the convolutional code, is a transparent code. This means that if the channel symbols have been inverted somewhere along the line, the decoders will still operate. The result will be the inversion of the original data. However, the Reed–Solomon code loses its transparency when the code is shortened. The "missing" bits in a shortened code need to be filled by either zeros or ones, depending on whether the data is complemented or not. (To put it another way, if the symbols are inverted, then the zero-fill needs to be inverted to a one-fill.) For this reason it is mandatory that the sense of the data (i.e., true or complemented) be resolved before Reed–Solomon decoding.
Designers are not required to use the "natural" sizes of Reed–Solomon code blocks. A technique known as "shortening" can produce a smaller code of any desired size from a larger code. For example, the widely used (255,223) code can be converted to a (160,128) code by padding the unused portion of the source block with 95 binary zeroes and not transmitting them. At the decoder, the same portion of the block is loaded locally with binary zeroes. The Delsarte–Goethals–Seidel theorem illustrates an example of an application of shortened Reed–Solomon codes. In parallel to shortening, a technique known as puncturing allows omitting some of the encoded parity symbols.
The decoders described in this section use the BCH view of a codeword as a sequence of coefficients. They use a fixed generator polynomial known to both encoder and decoder.
Daniel Gorenstein and Neal Zierler developed a decoder that was described in a MIT Lincoln Laboratory report by Zierler in January 1960 and later in a paper 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 (1961).
As a result of the Reed-Solomon encoding procedure, s(x) is divisible by the generator polynomial g(x):
Since s(x) is a multiple of the generator g(x), it follows that it "inherits" all its roots:
The transmitted polynomial is corrupted in transit by an error polynomial e(x) to produce the received polynomial r(x).
Coefficient ei will be zero if there is no error at that power of x and nonzero if there is an error. If there are ν errors at distinct powers ik of x, then
The goal of the decoder is to find the number of errors (ν), the positions of the errors (ik), and the error values at those positions (eik). From those, e(x) can be calculated and subtracted from r(x) to get the originally sent message s(x).
The advantage of looking at the syndromes is that the message polynomial drops out. In other words, the syndromes only relate to the error, and are unaffected by the actual contents of the message being transmitted. If the syndromes are all zero, the algorithm stops here and reports that the message was not corrupted in transit.
Then the syndromes can be written in terms of these error locators and error values as
The syndromes give a system of n − k ≥ 2ν equations in 2ν unknowns, but that system of equations is nonlinear in the Xk and does not have an obvious solution. However, if the Xk were known (see below), then the syndrome equations provide a linear system of equations that can easily be solved for the Yk error values.
Consequently, the problem is finding the Xk, because then the leftmost matrix would be known, and both sides of the equation could be multiplied by its inverse, yielding Yk
There is a linear recurrence relation that gives rise to a system of linear equations. Solving those equations identifies those error locations Xk.
These summations are now equivalent to the syndrome values, which we know and can substitute in! This therefore reduces to
Recall that j was chosen to be any integer between 1 and v inclusive, and this equivalence is true for any and all such values. Therefore, we have v linear equations, not just one. This system of linear equations can therefore be solved for the coefficients Λi of the error location polynomial:
The above assumes the decoder knows the number of errors ν, but that number has not been determined yet. The PGZ decoder does not determine ν directly but rather searches for it by trying successive values. The decoder first assumes the largest value for a trial ν and sets up the linear system for that value. If the equations can be solved (i.e., the matrix determinant is nonzero), then that trial value is the number of errors. If the linear system cannot be solved, then the trial ν is reduced by one and the next smaller system is examined. (Gill n.d., p. 35)
Finally, e(x) is generated from ik and eik and then is subtracted from r(x) to get the originally sent message s(x), with errors corrected.
Consider the Reed–Solomon code defined in GF(929) with α = 3 and t = 4 (this is used in PDF417 barcodes) for a RS(7,3) code. The generator polynomial is
If the message polynomial is p(x) = 3 x2 + 2 x + 1, then a systematic codeword is encoded as follows.
The coefficients can be reversed to produce roots with positive exponents, but typically this isn't used:
with the log of the roots corresponding to the error locations (right to left, location 0 is the last term in the codeword).
The Berlekamp–Massey algorithm is an alternate iterative procedure for finding the error locator polynomial. During each iteration, it calculates a discrepancy based on a current instance of Λ(x) with an assumed number of errors e:
and then adjusts Λ(x) and e so that a recalculated Δ would be zero. The article Berlekamp–Massey algorithm has a detailed description of the procedure. In the following example, C(x) is used to represent Λ(x).
Another iterative method for calculating both the error locator polynomial and the error value polynomial is based on Sugiyama's adaptation of the extended Euclidean algorithm .
The middle terms are zero due to the relationship between Λ and syndromes.
The extended Euclidean algorithm can find a series of polynomials of the form
where the degree of R decreases as i increases. Once the degree of Ri(x) < t/2, then
A discrete Fourier transform can be used for decoding. To avoid conflict with syndrome names, let c(x) = s(x) the encoded codeword. r(x) and e(x) are the same as above. Define C(x), E(x), and R(x) as the discrete Fourier transforms of c(x), e(x), and r(x). Since r(x) = c(x) + e(x), and since a discrete Fourier transform is a linear operator, R(x) = C(x) + E(x).
Transform r(x) to R(x) using discrete Fourier transform. Since the calculation for a discrete Fourier transform is the same as the calculation for syndromes, t coefficients of R(x) and E(x) are the same as the syndromes:
Then calculate C(x) = R(x) − E(x) and take the inverse transform (polynomial interpolation) of C(x) to produce c(x).
The Singleton bound states that the minimum distance d of a linear block code of size (n,k) is upper-bounded by n − k + 1. The distance d was usually understood to limit the error-correction capability to ⌊(d-1) / 2⌋. The Reed–Solomon code achieves this bound with equality, and can thus correct up to ⌊(n-k) / 2⌋ errors. However, this error-correction bound is not exact.
The algebraic decoding methods described above are hard-decision methods, which means that for every symbol a hard decision is made about its value. For example, a decoder could associate with each symbol an additional value corresponding to the channel demodulator's confidence in the correctness of the symbol. The advent of LDPC and turbo codes, which employ iterated soft-decision belief propagation decoding methods to achieve error-correction performance close to the theoretical limit, has spurred interest in applying soft-decision decoding to conventional algebraic codes. In 2003, Ralf Koetter and Alexander Vardy presented a polynomial-time soft-decision algebraic list-decoding algorithm for Reed–Solomon codes, which was based upon the work by Sudan and Guruswami. In 2016, Steven J. Franke and Joseph H. Taylor published a novel soft-decision decoder.% Multiply the information by X^(n-k), or just pad with zeros at the end to% Find the Reed-Solomon generating polynomial g(x), by the way this is the% Do the euclidean algorithm on the polynomials r0(x) and Syndromes(x) in% basically is nothing that we could do and we return the received message% Prepare a linear system to solve the error polynomial and find the error
The decoders described in this section use the Reed Solomon original view of a codeword as a sequence of polynomial values where the polynomial is based on the message to be encoded. The same set of fixed values are used by the encoder and decoder, and the decoder recovers the encoding polynomial (and optionally an error locating polynomial) from the received message.
In 1986, a decoder known as the Berlekamp–Welch algorithm was developed as a decoder that is able to recover the original message polynomial as well as an error "locator" polynomial that produces zeroes for the input values that correspond to errors, with time complexity O(n^3), where n is the number of values in a message. The recovered polynomial is then used to recover (recalculate as needed) the original message.
In 2002, an improved decoder was developed by Shuhong Gao, based on the extended Euclid algorithm .
divide Q(x) and E(x) by most significant coefficient of E(x) = 708. (Optional)