Reed–Solomon error correction

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 are a group of error-correcting codes that were introduced by Irving S. Reed and Gustave Solomon in 1960.^[1] They have many applications, the most prominent of which include consumer technologies such as CDs, DVDs, Blu-ray Discs, QR Codes, data transmission technologies such as DSL and WiMAX, broadcast systems such as DVB and ATSC, and storage systems such as RAID 6. They are also used in satellite communication.

Reed–Solomon codes operate on a block of data treated as a set of finite field elements called symbols. For example, a block of 4096 bytes (32768 bits) could be treated as a set of 2731 12 bit symbols, where each symbol is a finite field element of GF(2¹²), the last symbol padded with four 0 bits. Reed–Solomon codes are able to detect and correct multiple symbol errors. By adding $t$ check symbols to the data, a Reed–Solomon code can detect any combination of up to $t$ erroneous symbols, or correct up to $⌊ t /2⌋$ symbols. As an erasure code, it can correct up to $t$ known erasures, 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.

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". (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 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.^[2] The Gorenstein-Zierler decoder and the related work on BCH codes are described in a book Error Correcting Codes by W. Wesley Peterson (1961).^[3] 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,^[4], 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 is since 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.^[5]

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 type of decoders – see Guruswami–Sudan list decoding algorithm.

In 2002, another original scheme decoder was developed by Shuhong Gao, based on the extended Euclid algorithm Gao_RS.pdf .

Applications

Data storage

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.^[6]

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 to make use of Reed–Solomon when breaking up files.

Bar code

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.

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(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.

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 space probe.

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, being the Shannon capacity.

These concatenated codes are now being replaced by more powerful turbo codes.

Constructions

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.

Reed & Solomon's original view: The codeword as a sequence of values

There are different encoding procedures for the Reed–Solomon code, and thus, there are different ways to describe the set of all codewords. In the original view of Reed & Solomon (1960), every codeword of the Reed–Solomon code is a sequence of function values of a polynomial of degree less than k. In order to obtain a codeword of the Reed–Solomon code, the message is interpreted as the description of a polynomial p of degree less than k over the finite field F with q elements. In turn, the polynomial p is evaluated at n ≤ q distinct points $a_{1},\dots ,a_{n}$ of the field F, and the sequence of values is the corresponding codeword. Common choices for a set of evaluation points include {0, 1, 2, ..., n − 1}, {0, α, α², ..., αⁿ⁻², 1}, {1, α, α², ..., αⁿ⁻²}, ... , where α is a primitive element of F.

Formally, the set $\mathbf {C}$ of codewords of the Reed–Solomon code is defined as follows:

\mathbf {C} ={\Big \{}\;{\big (}p(a_{1}),p(a_{2}),\dots ,p(a_{n}){\big )}\;{\Big |}\;p{\text{ is a polynomial over }}F{\text{ of degree }}<k\;{\Big \}}\,.

Since any two distinct polynomials of degree less than $k$ agree in at most $k-1$ points, this means that any two codewords of the Reed–Solomon code disagree in at least $n-(k-1)=n-k+1$ positions. Furthermore, there are two polynomials that do agree in $k-2$ points but are not equal, and thus, the distance of the Reed–Solomon code is exactly $d=n-k+1$ . Then the relative distance is $\delta =d/n=1-k/n+1/n=1-R+1/n\sim 1-R$ , where $R=k/n$ is the rate. This trade-off between the relative distance and the rate is asymptotically optimal since, by the Singleton bound, every code satisfies $\delta +R\leq 1+1/n$ . Being a code that achieves this optimal trade-off, the Reed–Solomon code belongs to the class of maximum distance separable codes.

While the number of different polynomials of degree less than k and the number of different messages are both equal to $q^{k}$ , and thus every message can be uniquely mapped to such a polynomial, there are different ways of doing this encoding. The original construction of Reed & Solomon (1960) interprets the message x as the coefficients of the polynomial p, whereas subsequent constructions interpret the message as the values of the polynomial at the first k points $a_{1},\dots ,a_{k}$ and obtain the polynomial p by interpolating these values with a polynomial of degree less than k. The latter encoding procedure, while being slightly less efficient, has the advantage that it gives rise to a systematic code, that is, the original message is always contained as a subsequence of the codeword.

Simple encoding procedure: The message as a sequence of coefficients

In the original construction of Reed & Solomon (1960), the message $x=(x_{1},\dots ,x_{k})\in F^{k}$ is mapped to the polynomial $p_{x}$ with

p_{x}(a)=\sum _{i=1}^{k}x_{i}a^{i-1}\,.

The codeword of $x$ is obtained by evaluating $p_{x}$ at $n$ different points $a_{1},\dots ,a_{n}$ of the field $F$ . Thus the classical encoding function $C:F^{k}\to F^{n}$ for the Reed–Solomon code is defined as follows:

C(x)={\big (}p_{x}(a_{1}),\dots ,p_{x}(a_{n}){\big )}\,.

This function $C$ is a linear mapping, that is, it satisfies $C(x)=x\cdot A$ for the following $(k\times n)$ -matrix $A$ with elements from $F$ :

A={\begin{bmatrix}1&\dots &1&\dots &1\\a_{1}&\dots &a_{k}&\dots &a_{n}\\a_{1}^{2}&\dots &a_{k}^{2}&\dots &a_{n}^{2}\\\vdots &\dots &\vdots &\dots &\vdots \\a_{1}^{k-1}&\dots &a_{k}^{k-1}&\dots &a_{n}^{k-1}\end{bmatrix}}

This matrix is the transpose of a Vandermonde matrix over $F$ . In other words, the Reed–Solomon code is a linear code, and in the classical encoding procedure, its generator matrix is $A$ .

Systematic encoding procedure: The message as an initial sequence of values

There is an alternative encoding procedure that also produces the Reed–Solomon code, but that does so in a systematic way. Here, the mapping from the message $x$ to the polynomial $p_{x}$ works differently: the polynomial $p_{x}$ is now defined as the unique polynomial of degree less than $k$ such that

p_{x}(a_{i})=x_{i}

holds for all

i\in \{1,\dots ,k\}

.

To compute this polynomial $p_{x}$ from $x$ , one can use Lagrange interpolation. Once it has been found, it is evaluated at the other points $a_{k+1},\dots ,a_{n}$ of the field. The alternative encoding function $C:F^{k}\to F^{n}$ for the Reed–Solomon code is then again just the sequence of values:

C(x)={\big (}p_{x}(a_{1}),\dots ,p_{x}(a_{n}){\big )}\,.

Since the first $k$ entries of each codeword $C(x)$ coincide with $x$ , this encoding procedure is indeed systematic. Since Lagrange interpolation is a linear transformation, $C$ is a linear mapping. In fact, we have $C(x)=x\cdot G$ , where

G=(A{\text{'s left square submatrix}})^{-1}\cdot A={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}

Discrete Fourier transform and its inverse

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:

C(x)={\big (}p_{x}(a_{1}),\dots ,p_{x}(a_{n}){\big )}\,.

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:

a_{i}=\alpha ^{i-1}

a_{1},\dots ,a_{n}=\{1,\alpha ,\alpha ^{2},\dots ,\alpha ^{n-1}\}

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 BCH view: The codeword as a sequence of coefficients

In this view, the sender again maps the message $x$ to a polynomial $p_{x}$ , and for this, any of the two mappings just described can be used (where the message is either interpreted as the coefficients of $p_{x}$ or as the initial sequence of values of $p_{x}$ ). Once the sender has constructed the polynomial $p_{x}$ in some way, however, instead of sending the values of $p_{x}$ at all points, the sender computes some related polynomial $s$ of degree at most $n-1$ for $n=q-1$ and sends the $n$ coefficients of that polynomial. The polynomial $s(a)$ is constructed by multiplying the message polynomial $p_{x}(a)$ , which has degree at most $k-1$ , with a generator polynomial $g(a)$ of degree $n-k$ that is known to both the sender and the receiver. The generator polynomial $g(x)$ is defined as the polynomial whose roots are exactly $\alpha ,\alpha ^{2},\dots ,\alpha ^{n-k}$ , i.e.,

g(x)=(x-\alpha )(x-\alpha ^{2})\cdots (x-\alpha ^{n-k})=g_{0}+g_{1}x+\cdots +g_{n-k-1}x^{n-k-1}+x^{n-k}\,.

The transmitter sends the $n=q-1$ coefficients of $s(a)=p_{x}(a)\cdot g(a)$ . Thus, in the BCH view of Reed–Solomon codes, the set $\mathbf {C}$ of codewords is defined for $n=q-1$ as follows:^[7]

\mathbf {C} =\left\{\left(s_{1},s_{2},\dots ,s_{n}\right)\;{\Big |}\;s(a)=\sum _{i=1}^{n}s_{i}a^{i-1}{\text{ is a polynomial that has at least the roots }}\alpha ^{1},\alpha ^{2},\dots ,\alpha ^{n-k}\right\}\,.

Systematic encoding procedure

The encoding procedure for the BCH view of Reed–Solomon codes can be modified to yield a systematic encoding procedure, in which each codeword contains the message as a prefix. Here, instead of sending $s(x)=p(x)g(x)$ , the encoder constructs the transmitted polynomial $s(x)$ such that the coefficients of the $k$ largest monomials are equal to the corresponding coefficients of $p(x)$ , and the lower-order coefficients of $s(x)$ are chosen exactly in such a way that $s(x)$ becomes divisible by $g(x)$ . Then the coefficients of $p(x)$ are a subsequence of the coefficients of $s(x)$ . To get a code that is overall systematic, we construct the message polynomial $p(x)$ by interpreting the message as the sequence of its coefficients.

Formally, the construction is done by multiplying $p(x)$ by $x^{t}$ to make room for the $t=n-k$ check symbols, dividing that product by $g(x)$ to find the remainder, and then compensating for that remainder by subtracting it. The $t$ check symbols are created by computing the remainder $s_{r}(x)$ :

s_{r}(x)=p(x)\cdot x^{t}\ {\bmod {\ }}g(x).

Note that the remainder has degree at most $t-1$ , whereas the coefficients of $x^{t-1},x^{t-2},\dots ,x^{1},x^{0}$ in the polynomial $p(x)\cdot x^{t}$ are zero. Therefore, the following definition of the codeword $s(x)$ has the property that the first $k$ coefficients are identical to the coefficients of $p(x)$ :

s(x)=p(x)\cdot x^{t}-s_{r}(x)\,.

As a result, the codewords $s(x)$ are indeed elements of $\mathbf {C}$ , that is, they are divisible by the generator polynomial $g(x)$ :^[8]

s(x)\equiv p(x)\cdot x^{t}-s_{r}(x)\equiv s_{r}(x)-s_{r}(x)\equiv 0\mod g(x)\,.

Properties

The Reed–Solomon code is a [n, k, n − k + 1] code; in other words, it is a linear block code of length n (over F) with dimension k and minimum Hamming distance n − k + 1. The Reed–Solomon code is optimal in the sense that the minimum distance has the maximum value possible for a linear code of size (n, k); this is known as the Singleton bound. Such a code is also called a maximum distance separable (MDS) code.

The error-correcting ability of a Reed–Solomon code is determined by its minimum distance, or equivalently, by $n-k$ , the measure of redundancy in the block. If the locations of the error symbols are not known in advance, then a Reed–Solomon code can correct up to $(n-k)/2$ erroneous symbols, i.e., it can correct half as many errors as there are redundant symbols added to the block. Sometimes error locations are known in advance (e.g., "side information" in demodulator signal-to-noise ratios)—these are called erasures. A Reed–Solomon code (like any MDS code) is able to correct twice as many erasures as errors, and any combination of errors and erasures can be corrected as long as the relation 2E + S ≤ n − k is satisfied, where $E$ is the number of errors and $S$ is the number of erasures in the block.

For practical uses of Reed–Solomon codes, it is common to use a finite field $F$ with $2^{m}$ elements. In this case, each symbol can be represented as an $m$ -bit value. The sender sends the data points as encoded blocks, and the number of symbols in the encoded block is $n=2^{m}-1$ . Thus a Reed–Solomon code operating on 8-bit symbols has $n=2^{8}-1=255$ symbols per block. (This is a very popular value because of the prevalence of byte-oriented computer systems.) The number $k$ , with $k<n$ , of data symbols in the block is a design parameter. A commonly used code encodes $k=223$ eight-bit data symbols plus 32 eight-bit parity symbols in an $n=255$ -symbol block; this is denoted as a $(n,k)=(255,223)$ code, and is capable of correcting up to 16 symbol errors per block.

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.

Whether the Reed–Solomon code is cyclic or not depends on subtle details of the construction. In the original view of Reed and Solomon, where the codewords are the values of a polynomial, one can choose the sequence of evaluation points in such a way as to make the code cyclic. In particular, if $\alpha$ is a primitive root of the field $F$ , then by definition all non-zero elements of $F$ take the form $\alpha ^{i}$ for $i\in \{1,\dots ,q-1\}$ , where $q=|F|$ . Each polynomial $p$ over $F$ gives rise to a codeword $(p(\alpha ^{1}),\dots ,p(\alpha ^{q-1}))$ . Since the function $a\mapsto p(\alpha a)$ is also a polynomial of the same degree, this function gives rise to a codeword $(p(\alpha ^{2}),\dots ,p(\alpha ^{q}))$ ; since $\alpha ^{q}=\alpha ^{1}$ holds, this codeword is the cyclic left-shift of the original codeword derived from $p$ . So choosing a sequence of primitive root powers as the evaluation points makes the original view Reed–Solomon code cyclic. Reed–Solomon codes in the BCH view are always cyclic because BCH codes are cyclic.

Remarks

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^[9] 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.

Reed Solomon original view decoders

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.

Theoretical decoder

Reed & Solomon (1960) described a theoretical decoder that corrected errors by finding the most popular message polynomial. The decoder only knows the set of values $a_{1}$ to $a_{n}$ and which encoding method was used to generate the codeword's sequence of values. The original message, the polynomial, and any errors are unknown. A decoding procedure could use a method like Lagrange interpolation on various subsets of n codeword values taken k at a time to repeatedly produce potential polynomials, until a sufficient number of matching polynomials are produced to reasonably eliminate any errors in the received codeword. Once a polynomial is determined, then any errors in the codeword can be corrected, by recalculating the corresponding codeword values. Unfortunately, in all but the simplest of cases, there are too many subsets, so the algorithm is impractical. The number of subsets is the binomial coefficient, $\textstyle {\binom {n}{k}}={n! \over (n-k)!k!}$ , and the number of subsets is infeasible for even modest codes. For a $(255,249)$ code that can correct 3 errors, the naive theoretical decoder would examine 359 billion subsets.

Berlekamp Welch decoder

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.

Example

Using RS(7,3), GF(929), and the set of evaluation points a_i = i-1

a = {0, 1, 2, 3, 4, 5, 6}

If the message polynomial is

p (x) = 003 x 2 + 002 x + 001

The codeword is

c = {001, 006, 017, 034, 057, 086, 121}

Errors in transmission might cause this to be received instead.

b = c + e = {001, 006, 123, 456, 057, 086, 121}

The key equations are:

b_{i}E(a_{i})-Q(a_{i})=0

Assume maximum number of errors: e = 2. The key equations become:

b_{i}(e_{0}+e_{1}a_{i})-(q_{0}+q_{1}a_{i}+q_{2}a_{i}^{2}+q_{3}a_{i}^{3}+q_{4}a_{i}^{4})=-b_{i}a_{i}^{2}

{\begin{bmatrix}001&000&928&000&000&000&000\\006&006&928&928&928&928&928\\123&246&928&927&925&921&913\\456&439&928&926&920&902&848\\057&228&928&925&913&865&673\\086&430&928&924&904&804&304\\121&726&928&923&893&713&562\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q_{0}\\q_{1}\\q_{2}\\q_{3}\\q_{4}\end{bmatrix}}={\begin{bmatrix}000\\923\\437\\541\\017\\637\\289\end{bmatrix}}

Using Gaussian elimination:

{\begin{bmatrix}001&000&000&000&000&000&000\\000&001&000&000&000&000&000\\000&000&001&000&000&000&000\\000&000&000&001&000&000&000\\000&000&000&000&001&000&000\\000&000&000&000&000&001&000\\000&000&000&000&000&000&001\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q_{0}\\q_{1}\\q_{2}\\q_{3}\\q_{4}\end{bmatrix}}={\begin{bmatrix}006\\924\\006\\007\\009\\916\\003\end{bmatrix}}

Q (x) = 003 x 4 + 916 x 3 + 009 x 2 + 007 x + 006

E (x) = 001 x 2 + 924 x + 006

Q (x) / E (x) = P (x) = 003 x 2 + 002 x + 001

Recalculate $P(x)$ where $E(x) = 0 : {2, 3}$ to correct $b$ resulting in the corrected codeword:

c = {001, 006, 017, 034, 057, 086, 121}

Gao decoder

In 2002, an improved decoder was developed by Shuhong Gao, based on the extended Euclid algorithm Gao_RS.pdf .

Example

Using the same data as the Berlekamp Welch example above:

R_{-1}=\prod _{i=1}^{n}(x-a_{i})

R_{0}=

Lagrange interpolation of

\{a_{i},b(a_{i})\}

for i = 1 to n

A_{-1}=0

A_{0}=1

i	R_i	A_i
−1	001 x⁷ + 908 x⁶ + 175 x⁵ + 194 x⁴ + 695 x³ + 094 x² + 720 x + 000	000
0	055 x⁶ + 440 x⁵ + 497 x⁴ + 904 x³ + 424 x² + 472 x + 001	001
1	702 x⁵ + 845 x⁴ + 691 x³ + 461 x² + 327 x + 237	152 x + 237
2	266 x⁴ + 086 x³ + 798 x² + 311 x + 532	708 x² + 176 x + 532

Q (x) = R 2 = 266 x 4 + 086 x 3 + 798 x 2 + 311 x + 532

E (x) = A 2 = 708 x 2 + 176 x + 532

divide Q(x) and E(x) by most significant coeficient of E(x) = 708. (Optional)

Q (x) = 003 x 4 + 916 x 3 + 009 x 2 + 007 x + 006

E (x) = 001 x 2 + 924 x + 006

Q (x) / E (x) = P (x) = 003 x 2 + 002 x + 001

Recalculate $P(x)$ where $E(x) = 0 : {2, 3}$ to correct $b$ resulting in the corrected codeword:

c = {001, 006, 017, 034, 057, 086, 121}

BCH view decoders

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.

Peterson–Gorenstein–Zierler 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.^[10] The Gorenstein-Zierler decoder and the related work on BCH codes are described in a book Error Correcting Codes by W. Wesley Peterson (1961).^[11]

Syndrome decoding

The transmitted message is viewed as the coefficients of a polynomial s(x) that is divisible by a generator polynomial g(x).

s(x)=\sum _{i=0}^{n-1}c_{i}x^{i}

g(x)=\prod _{j=1}^{n-k}(x-\alpha ^{j}),

where α is a primitive root.

Since s(x) is divisible by generator g(x), it follows that

s(\alpha ^{i})=0,\ i=1,2,\ldots ,n-k

The transmitted polynomial is corrupted in transit by an error polynomial e(x) to produce the received polynomial r(x).

r(x)=s(x)+e(x)

e(x)=\sum _{i=0}^{n-1}e_{i}x^{i}

where e_i is the coefficient for the i-th power of x. Coefficient e_i 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 i_k of x, then

e(x)=\sum _{k=1}^{\nu }e_{i_{k}}x^{i_{k}}

The goal of the decoder is to find the number of errors (ν), the positions of the errors (i_k), and the error values at those positions (e_{i_k}). From those, e(x) can be calculated and subtracted from r(x) to get the original message s(x).

The syndromes S_j are defined as

{\begin{aligned}S_{j}&=r(\alpha ^{j})=s(\alpha ^{j})+e(\alpha ^{j})=0+e(\alpha ^{j})=e(\alpha ^{j}),\ j=1,2,\ldots ,n-k\\&=\sum _{k=1}^{\nu }e_{i_{k}}\left(\alpha ^{j}\right)^{i_{k}}\end{aligned}}

The advantage of looking at the syndromes is that the message polynomial drops out.

Error locators and error values

For convenience, define the error locators X_k and error values Y_k as:

X_{k}=\alpha ^{i_{k}},\ Y_{k}=e_{i_{k}}

Then the syndromes can be written in terms of the error locators and error values as

S_{j}=\sum _{k=1}^{\nu }Y_{k}X_{k}^{j}

The syndromes give a system of n − k ≥ 2ν equations in 2ν unknowns, but that system of equations is nonlinear in the X_k and does not have an obvious solution. However, if the X_k were known (see below), then the syndrome equations provide a linear system of equations that can easily be solved for the Y_k error values.

{\begin{bmatrix}X_{1}^{1}&X_{2}^{1}&\cdots &X_{\nu }^{1}\\X_{1}^{2}&X_{2}^{2}&\cdots &X_{\nu }^{2}\\\vdots &\vdots &&\vdots \\X_{1}^{n-k}&X_{2}^{n-k}&\cdots &X_{\nu }^{n-k}\\\end{bmatrix}}{\begin{bmatrix}Y_{1}\\Y_{2}\\\vdots \\Y_{\nu }\end{bmatrix}}={\begin{bmatrix}S_{1}\\S_{2}\\\vdots \\S_{n-k}\end{bmatrix}}

Consequently, the problem is finding the X_k, because then the leftmost matrix would be known, and both sides of the equation could be multiplied by its inverse, yielding Y_k

Error locator polynomial

There is a linear recurrence relation that gives rise to a system of linear equations. Solving those equations identifies the error locations.

Define the error locator polynomial Λ(x) as

\Lambda (x)=\prod _{k=1}^{\nu }(1-xX_{k})=1+\Lambda _{1}x^{1}+\Lambda _{2}x^{2}+\cdots +\Lambda _{\nu }x^{\nu }

The zeros of Λ(x) are the reciprocals $X_{k}^{-1}$ :

\Lambda (X_{k}^{-1})=0

\Lambda (X_{k}^{-1})=1+\Lambda _{1}X_{k}^{-1}+\Lambda _{2}X_{k}^{-2}+\cdots +\Lambda _{\nu }X_{k}^{-\nu }=0

Multiply both sides by $Y_{k}X_{k}^{j+\nu }$ and it will still be zero. j is any number such that 1≤j≤v.

{\begin{aligned}&Y_{k}X_{k}^{j+\nu }\Lambda (X_{k}^{-1})=0.\\{\text{Hence }}&Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu }X_{k}^{-1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu }X_{k}^{-2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j+\nu }X_{k}^{-\nu }=0,\\{\text{and so }}&Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j}=0\\\end{aligned}}

Sum for k = 1 to ν

{\begin{aligned}&\sum _{k=1}^{\nu }(Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j})=0\\&\sum _{k=1}^{\nu }(Y_{k}X_{k}^{j+\nu })+\Lambda _{1}\sum _{k=1}^{\nu }(Y_{k}X_{k}^{j+\nu -1})+\Lambda _{2}\sum _{k=1}^{\nu }(Y_{k}X_{k}^{j+\nu -2})+\cdots +\Lambda _{\nu }\sum _{k=1}^{\nu }(Y_{k}X_{k}^{j})=0\end{aligned}}

This reduces to

S_{j+\nu }+\Lambda _{1}S_{j+\nu -1}+\cdots +\Lambda _{\nu -1}S_{j+1}+\Lambda _{\nu }S_{j}=0\,

S_{j}\Lambda _{\nu }+S_{j+1}\Lambda _{\nu -1}+\cdots +S_{j+\nu -1}\Lambda _{1}=-S_{j+\nu }\

This yields a system of linear equations that can be solved for the coefficients Λ_i of the error location polynomial:

{\begin{bmatrix}S_{1}&S_{2}&\cdots &S_{\nu }\\S_{2}&S_{3}&\cdots &S_{\nu +1}\\\vdots &\vdots &&\vdots \\S_{\nu }&S_{\nu +1}&\cdots &S_{2\nu -1}\end{bmatrix}}{\begin{bmatrix}\Lambda _{\nu }\\\Lambda _{\nu -1}\\\vdots \\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}-S_{\nu +1}\\-S_{\nu +2}\\\vdots \\-S_{\nu +\nu }\end{bmatrix}}

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)

Obtain the error locators from the error locator polynomial

Use the coefficients Λ_i found in the last step to build the error location polynomial. The roots of the error location polynomial can be found by exhaustive search. The error locators are the reciprocals of those roots. Note that the order of coefficients of the error location polynomial can be reversed, in which case the roots of that polynomial are the error locators (not reciprocals). Chien search is an efficient implementation of this step.

Calculate the error locations

Calculate i_k by taking the log base a of X_k. This is generally done using a precomputed lookup table.

Calculate the error values

Once the error locators are known, the error values can be determined. This can be done by direct solution for Y_k in the error equations given above, or using the Forney algorithm.

Fix the errors

Finally, e(x) is generated from i_k and e_{i_k} and then is subtracted from r(x) to get the sent message s(x).

Example

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

g(x)=(x-3)(x-3^{2})(x-3^{3})(x-3^{4})=x^{4}+809x^{3}+723x^{2}+568x+522

If the message polynomial is $p (x) = 3 x 2 + 2 x + 1$ , then a systematic codeword is encoded as follows.

s_{r}(x)=p(x)\,x^{t}\mod g(x)=547x^{3}+738x^{2}+442x+455

s(x)=p(x)\,x^{t}-s_{r}(x)=3x^{6}+2x^{5}+1x^{4}+382x^{3}+191x^{2}+487x+474

Errors in transmission might cause this to be received instead.

r(x)=s(x)+e(x)=3x^{6}+2x^{5}+123x^{4}+456x^{3}+191x^{2}+487x+474

The syndromes are calculated by evaluating r at powers of α.

S_{1}=r(3^{1})=3\cdot 3^{6}+2\cdot 3^{5}+123\cdot 3^{4}+456\cdot 3^{3}+191\cdot 3^{2}+487\cdot 3+474=732

S_{2}=r(3^{2})=637,\;S_{3}=r(3^{3})=762,\;S_{4}=r(3^{4})=925

{\begin{bmatrix}732&637\\637&762\end{bmatrix}}{\begin{bmatrix}\Lambda _{2}\\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}-762\\-925\end{bmatrix}}={\begin{bmatrix}167\\004\end{bmatrix}}

Using Gaussian elimination:

{\begin{bmatrix}001&000\\000&001\end{bmatrix}}{\begin{bmatrix}\Lambda _{2}\\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}329\\821\end{bmatrix}}

Λ(x) = 329 x² + 821 x + 001, with roots x₁ = 757 = 3⁻³ and x₂ = 562 = 3⁻⁴

The coefficients can be reversed to produce roots with positive exponents, but typically this isn't used:

R(x) = 001 x² + 821 x + 329, with roots 27 = 3³ and 81 = 3⁴

with the log of the roots corresponding to the error locations (right to left, location 0 is the last term in the codeword).

To calculate the error values, apply the Forney algorithm.

Ω(x) = S(x) Λ(x) mod x⁴ = 546 x + 732

Λ'(x) = 658 x + 821

e₁ = -Ω(x₁)/Λ'(x₁) = 074

e₂ = -Ω(x₂)/Λ'(x₂) = 122

Subtracting e₁ x³ and e₂ x⁴ from the received polynomial r reproduces the original codeword s.

Berlekamp–Massey decoder

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:

\Delta =S_{i}+\Lambda _{1}\ S_{i-1}+\cdots +\Lambda _{e}\ S_{i-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).

Example

Using the same data as the Peterson Gorenstein Zierler example above:

n	S_n+1	d	C	B	b	m
0	732	732	197 x + 1	1	732	1
1	637	846	173 x + 1	1	732	2
2	762	412	634 x² + 173 x + 1	173 x + 1	412	1
3	925	576	329 x² + 821 x + 1	173 x + 1	412	2

The final value of C is the error locator polynomial, Λ(x).

Euclidean decoder

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 .

Define S(x), Λ(x), and Ω(x) for t syndromes and e errors:

S(x)=S_{t}x^{t-1}+S_{t-1}x^{t-2}+\cdots +S_{2}x+S_{1}

\Lambda (x)=\Lambda _{e}x^{e}+\Lambda _{e-1}x^{e-1}+\cdots +\Lambda _{1}x+1

\Omega (x)=\Omega _{e}x^{e}+\Omega _{e-1}x^{e-1}+\cdots +\Omega _{1}x+\Omega _{0}

The key equation is:

\Lambda (x)S(x)=Q(x)x^{t}+\Omega (x)

For t = 6 and e = 3:

{\begin{bmatrix}\Lambda _{3}S_{6}&x^{8}\\\Lambda _{2}S_{6}+\Lambda _{3}S_{5}&x^{7}\\\Lambda _{1}S_{6}+\Lambda _{2}S_{5}+\Lambda _{3}S_{4}&x^{6}\\S_{6}+\Lambda _{1}S_{5}+\Lambda _{2}S_{4}+\Lambda _{3}S_{3}&x^{5}\\S_{5}+\Lambda _{1}S_{4}+\Lambda _{2}S_{3}+\Lambda _{3}S_{2}&x^{4}\\S_{4}+\Lambda _{1}S_{3}+\Lambda _{2}S_{2}+\Lambda _{3}S_{1}&x^{3}\\S_{3}+\Lambda _{1}S_{2}+\Lambda _{2}S_{1}&x^{2}\\S_{2}+\Lambda _{1}S_{1}&x\\S_{1}&\\\end{bmatrix}}={\begin{bmatrix}Q_{2}x^{8}\\Q_{1}x^{7}\\Q_{0}x^{6}\\0\\0\\0\\\Omega _{2}x^{2}\\\Omega _{1}x\\\Omega _{0}\\\end{bmatrix}}

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

A_i(x) S(x) + B_i(x) x^t = R_i(x)

where the degree of R decreases as i increases. Once the degree of R_i(x) < t/2, then

A_i(x) = Λ(x)

B_i(x) = -Q(x)

R_i(x) = Ω(x).

B(x) and Q(x) don't need to be saved, so the algorithm becomes:

R₋₁ = x^t

R₀ = S(x)

A₋₁ = 0

A₀ = 1

i = 0

while degree of R_i >= t/2

i = i + 1

Q = R_i-2 / R_i-1

R_i = R_i-2 - Q R_i-1

A_i = A_i-2 - Q A_i-1

to set low order term of Λ(x) to 1, divide Λ(x) and Ω(x) by A_i(0):

Λ(x) = A_i / A_i(0)

Ω(x) = R_i / A_i(0)

A_i(0) is the constant (low order) term of A_i.

Example

Using the same data as the Peterson Gorenstein Zierler example above:

i	R_i	A_i
-1	001 x⁴ + 000 x³ + 000 x² + 000 x + 000	000
0	925 x³ + 762 x² + 637 x + 732	001
1	683 x² + 676 x + 024	697 x + 396
2	673 x + 596	608 x² + 704 x + 544

Λ(x) = A₂ / 544 = 329 x² + 821 x + 001

Ω(x) = R₂ / 544 = 546 x + 732

Decoder using discrete Fourier transform

A discrete Fourier transform can be used for decoding.^[12] 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:

R_{j}=E_{j}=S_{j}=r(\alpha ^{j})

for\ 1\leq j\leq t

Use $R_{1}$ through $R_{t}$ as syndromes (they're the same) and generate the error locator polynomial using the methods from any of the above decoders.

Let v = number of errors. Generate E(x) using the known coefficients $E_{1}$ to $E_{t}$ , the error locator polynomial, and these formulas

E_{0}=-{\frac {1}{\Lambda _{v}}}(E_{v}+\Lambda _{1}E_{v-1}+\cdots +\Lambda _{v-1}E_{1})

E_{j}=-(\Lambda _{1}E_{j-1}+\Lambda _{2}E_{j-2}+\cdots +\Lambda _{v}E_{j-v})

for\ t<j<n

Then calculate C(x) = R(x) - E(x) and take the inverse transform (polynomial interpolation) of C(x) to produce c(x).

Decoding beyond the error-correction bound

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/2⌋. The Reed–Solomon code achieves this bound with equality, and can thus correct up to ⌊(n − k + 1)/2⌋ errors. However, this error-correction bound is not exact.

In 1999, Madhu Sudan and Venkatesan Guruswami at MIT published "Improved Decoding of Reed–Solomon and Algebraic-Geometry Codes" introducing an algorithm that allowed for the correction of errors beyond half the minimum distance of the code.^[13] It applies to Reed–Solomon codes and more generally to algebraic geometric codes. This algorithm produces a list of codewords (it is a list-decoding algorithm) and is based on interpolation and factorization of polynomials over $GF(2^{m})$ and its extensions.

Soft-decoding

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.^[14] In 2016, Steven J. Franke and Joseph H. Taylor published a novel soft-decision decoder.^[15]

Matlab Example

Encoder

Here we present a simple Matlab implementation for an encoder.

function [ encoded ] = rsEncoder( msg, m, prim_poly, n, k )
    %RSENCODER Encode message with the Reed-Solomon algorithm
    % m is the number of bits per symbol
    % prim_poly: Primitive polynomial p(x). Ie for DM is 301
    % k is the size of the message
    % n is the total size (k+redundant)
    % Example: msg = uint8('Test')
    % enc_msg = rsEncoder(msg, 8, 301, 12, numel(msg));
    
    % Get the alpha
    alpha = gf(2, m, prim_poly);
    
    % Get the Reed-Solomon generating polynomial g(x)
    g_x = genpoly(k, n, alpha);
    
    % Multiply the information by X^(n-k), or just pad with zeros at the end to
    % get space to add the redundant information
    msg_padded = gf([msg zeros(1, n-k)], m, prim_poly);
    
    % Get the remainder of the division of the extended message by the 
    % Reed-Solomon generating polynomial g(x)
    [~, remainder] = deconv(msg_padded, g_x);

    % Now return the message with the redundant information
    encoded = msg_padded - remainder;

end

% Find the Reed-Solomon generating polynomial g(x), by the way this is the
% same as the rsgenpoly function on matlab
function g = genpoly(k, n, alpha)
    g = 1;
    % A multiplication on the galois field is just a convolution
    for k = mod(1 : n-k, n)
        g = conv(g, [1 alpha .^ (k)]);
    end
end

Decoder

Now the decoding part:

function [ decoded, error_pos, error_mag, g, S ] = rsDecoder( encoded, m, prim_poly, n, k )
    %RSDECODER Decode a Reed-Solomon encoded message
    %   Example:
    % [dec, ~, ~, ~, ~] = rsDecoder(enc_msg, 8, 301, 12, numel(msg))
    max_errors = floor((n-k)/2);
    orig_vals = encoded.x;
    % Initialize the error vector
    errors = zeros(1, n);
    g = [];
    S = [];
    
    % Get the alpha
    alpha = gf(2, m, prim_poly);
    
    % Find the syndromes (Check if dividing the message by the generator
    % polynomial the result is zero)
    Synd = polyval(encoded, alpha .^ (1:n-k));
    Syndromes = trim(Synd);
    
    % If all syndromes are zeros (perfectly divisible) there are no errors
    if isempty(Syndromes.x)
        decoded = orig_vals(1:k);
        error_pos = [];
        error_mag = [];
        g = [];
        S = Synd;
        return;
    end
    
    % Prepare for the euclidean algorithm (Used to find the error locating
    % polynomials)
    r0 = [1, zeros(1, 2*max_errors)]; r0 = gf(r0, m, prim_poly); r0 = trim(r0);
    size_r0 = length(r0);
    r1 = Syndromes;
    f0 = gf([zeros(1, size_r0-1) 1], m, prim_poly);
    f1 = gf(zeros(1, size_r0), m, prim_poly);
    g0 = f1; g1 = f0;
    
    % Do the euclidean algorithm on the polynomials r0(x) and Syndromes(x) in
    % order to find the error locating polynomial
    while true
        % Do a long division
        [quotient, remainder] = deconv(r0, r1);    
        % Add some zeros
        quotient = pad(quotient, length(g1));
        
        % Find quotient*g1 and pad
        c = conv(quotient, g1);
        c = trim(c);
        c = pad(c, length(g0));
        
        % Update g as g0-quotient*g1
        g = g0 - c;
        
        % Check if the degree of remainder(x) is less than max_errors
        if all(remainder(1:end - max_errors) == 0)
            break;
        end
        
        % Update r0, r1, g0, g1 and remove leading zeros
        r0 = trim(r1); r1 = trim(remainder);
        g0 = g1; g1 = g;
    end
    
    % Remove leading zeros
    g = trim(g);
    
    % Find the zeros of the error polynomial on this galois field
    evalPoly = polyval(g, alpha .^ (n-1 : -1 : 0));
    error_pos = gf(find(evalPoly == 0), m);
    
    % If no error position is found we return the received work, because
    % basically is nothing that we could do and we return the received message
    if isempty(error_pos)
        decoded = orig_vals(1:k);
        error_mag = [];
        return;
    end
    
    % Prepare a linear system to solve the error polynomial and find the error
    % magnitudes
    size_error = length(error_pos);
    Syndrome_Vals = Syndromes.x;
    b(:, 1) = Syndrome_Vals(1:size_error);
    for idx = 1 : size_error
        e = alpha .^ (idx*(n-error_pos.x));
        err = e.x;
        er(idx, :) = err;
    end
    
    % Solve the linear system
    error_mag = (gf(er, m, prim_poly) \ gf(b, m, prim_poly))';
    % Put the error magnitude on the error vector
    errors(error_pos.x) = error_mag.x;
    % Bring this vector to the galois field
    errors_gf = gf(errors, m, prim_poly);
    
    % Now to fix the errors just add with the encoded code
    decoded_gf = encoded(1:k) + errors_gf(1:k);
    decoded = decoded_gf.x;
    
end

% Remove leading zeros from galois array
function gt = trim(g)
    gx = g.x;
    gt = gf(gx(find(gx, 1) : end), g.m, g.prim_poly);
end

% Add leading zeros
function xpad = pad(x,k)
    len = length(x);
    if (len<k)
        xpad = [zeros(1, k-len) x];
    end
end

Notes

↑ Reed & Solomon (1960)
↑ D. Gorenstein and N. Zierler, "A class of cyclic linear error-correcting codes in p^m symbols", J. SIAM, vol. 9, pp. 207-214, June 1961
↑ Error Correcting Codes by W_Wesley_Peterson, 1961
↑ Error Correcting Codes by W_Wesley_Peterson, second edition, 1972
↑ Yasuo Sugiyama, Masao Kasahara, Shigeichi Hirasawa, and Toshihiko Namekawa. A method for solving key equation for decoding Goppa codes. Information and Control, 27:87–99, 1975.
↑ Immink, K. A. S. (1994), "Reed–Solomon Codes and the Compact Disc", in Wicker, Stephen B.; Bhargava, Vijay K., Reed–Solomon Codes and Their Applications, IEEE Press, ISBN 978-0-7803-1025-4
↑ Lidl, Rudolf; Pilz, Günter (1999). Applied Abstract Algebra (2nd ed.). Wiley. p. 226.
↑ See Lin & Costello (1983, p. 171), for example.
↑ Pfender, Florian; Ziegler, Günter M. (September 2004), "Kissing Numbers, Sphere Packings, and Some Unexpected Proofs" (PDF), Notices of the American Mathematical Society, 51 (8): 873&ndash, 883 . Explains the Delsarte-Goethals-Seidel theorem as used in the context of the error correcting code for compact disc.
↑ D. Gorenstein and N. Zierler, "A class of cyclic linear error-correcting codes in p^m symbols," J. SIAM, vol. 9, pp. 207-214, June 1961
↑ Error Correcting Codes by W_Wesley_Peterson, 1961
↑ Shu Lin and Daniel J. Costello Jr, "Error Control Coding" second edition, pp. 255-262, 1982, 2004
↑ Guruswami, V.; Sudan, M. (September 1999), "Improved decoding of Reed–Solomon codes and algebraic geometry codes", IEEE Transactions on Information Theory, 45 (6): 1757&ndash, 1767, doi:10.1109/18.782097
↑ Koetter, Ralf; Vardy, Alexander (2003). "Algebraic soft-decision decoding of Reed–Solomon codes". IEEE Transactions on Information Theory. 49 (11): 2809–2825. doi:10.1109/TIT.2003.819332.
↑ Franke, Steven J.; Taylor, Joseph H. (2016). "Open Source Soft-Decision Decoder for the JT65 (63,12) Reed–Solomon Code" (PDF). QEX (May/June): 8–17.

References

Gill, John (n.d.), EE387 Notes #7, Handout #28 (PDF), Stanford University, archived from the original (PDF) on June 30, 2014, retrieved April 21, 2010
Hong, Jonathan; Vetterli, Martin (August 1995), "Simple Algorithms for BCH Decoding" (PDF), IEEE Transactions on Communications, 43 (8): 2324&ndash, 2333, doi:10.1109/26.403765
Lin, Shu; Costello, Jr., Daniel J. (1983), Error Control Coding: Fundamentals and Applications, New Jersey, NJ: Prentice-Hall, ISBN 0-13-283796-X
Massey, J. L. (1969), "Shift-register synthesis and BCH decoding" (PDF), IEEE Transactions on Information Theory, IT-15 (1): 122–127, doi:10.1109/tit.1969.1054260
Peterson, Wesley W. (1960), "Encoding and Error Correction Procedures for the Bose-Chaudhuri Codes", IRE Transactions on Information Theory, Institute of Radio Engineers, IT-6: 459–470
Reed, Irving S.; Solomon, Gustave (1960), "Polynomial Codes over Certain Finite Fields", Journal of the Society for Industrial and Applied Mathematics (SIAM), 8 (2): 300–304, doi:10.1137/0108018
Welch, L. R. (1997), The Original View of Reed–Solomon Codes (PDF), Lecture Notes

External links

Information and Tutorials

Introduction to Reed–Solomon codes: principles, architecture and implementation (CMU)
A Tutorial on Reed–Solomon Coding for Fault-Tolerance in RAID-like Systems
Algebraic soft-decoding of Reed–Solomon codes
Wikiversity: Reed–Solomon codes for coders
BBC R&D White Paper WHP031
Geisel, William A. (August 1990), Tutorial on Reed–Solomon Error Correction Coding (PDF), Technical Memorandum, NASA, TM-102162
Gao, Shuhong (January 2002), New Algorithm For Decoding Reed-Solomon Codes (PDF), Clemson

Implementations

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Reed & Solomon (1960)

[2] D. Gorenstein and N. Zierler, "A class of cyclic linear error-correcting codes in p^m symbols", J. SIAM, vol. 9, pp. 207-214, June 1961

[3] Error Correcting Codes by W_Wesley_Peterson, 1961

[4] Error Correcting Codes by W_Wesley_Peterson, second edition, 1972

[5] Yasuo Sugiyama, Masao Kasahara, Shigeichi Hirasawa, and Toshihiko Namekawa. A method for solving key equation for decoding Goppa codes. Information and Control, 27:87–99, 1975.

[6] Immink, K. A. S. (1994), "Reed–Solomon Codes and the Compact Disc", in Wicker, Stephen B.; Bhargava, Vijay K., Reed–Solomon Codes and Their Applications, IEEE Press, ISBN 978-0-7803-1025-4

[7] Lidl, Rudolf; Pilz, Günter (1999). Applied Abstract Algebra (2nd ed.). Wiley. p. 226.

[8] See Lin & Costello (1983, p. 171), for example.

[9] Pfender, Florian; Ziegler, Günter M. (September 2004), "Kissing Numbers, Sphere Packings, and Some Unexpected Proofs" (PDF), Notices of the American Mathematical Society, 51 (8): 873&ndash, 883 . Explains the Delsarte-Goethals-Seidel theorem as used in the context of the error correcting code for compact disc.

[10] D. Gorenstein and N. Zierler, "A class of cyclic linear error-correcting codes in p^m symbols," J. SIAM, vol. 9, pp. 207-214, June 1961

[11] Error Correcting Codes by W_Wesley_Peterson, 1961

[12] Shu Lin and Daniel J. Costello Jr, "Error Control Coding" second edition, pp. 255-262, 1982, 2004

[13] Guruswami, V.; Sudan, M. (September 1999), "Improved decoding of Reed–Solomon codes and algebraic geometry codes", IEEE Transactions on Information Theory, 45 (6): 1757&ndash, 1767, doi:10.1109/18.782097

[14] Koetter, Ralf; Vardy, Alexander (2003). "Algebraic soft-decision decoding of Reed–Solomon codes". IEEE Transactions on Information Theory. 49 (11): 2809–2825. doi:10.1109/TIT.2003.819332.

[15] Franke, Steven J.; Taylor, Joseph H. (2016). "Open Source Soft-Decision Decoder for the JT65 (63,12) Reed–Solomon Code" (PDF). QEX (May/June): 8–17.

Reed–Solomon error correction

History

Applications

Data storage

Bar code

Data transmission

Space transmission

Constructions

Reed & Solomon's original view: The codeword as a sequence of values

Simple encoding procedure: The message as a sequence of coefficients

Systematic encoding procedure: The message as an initial sequence of values

Discrete Fourier transform and its inverse

The BCH view: The codeword as a sequence of coefficients

Systematic encoding procedure

Properties

Remarks

Reed Solomon original view decoders

Theoretical decoder

Berlekamp Welch decoder

Example

Gao decoder

Example

BCH view decoders

Peterson–Gorenstein–Zierler decoder

Syndrome decoding

Error locators and error values

Error locator polynomial

Obtain the error locators from the error locator polynomial

Calculate the error locations

Calculate the error values

Fix the errors

Example

Berlekamp–Massey decoder

Example

Euclidean decoder

Example

Decoder using discrete Fourier transform

Decoding beyond the error-correction bound

Soft-decoding

Matlab Example

Encoder

Decoder

See also

Notes

References

Further reading

External links

Information and Tutorials

Implementations