information theory Information theory is the mathematical study of the quantification (science), quantification, Data storage, storage, and telecommunications, communication of information. The field was established and formalized by Claude Shannon in the 1940s, ...

and

coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and computer data storage, data sto ...

, Reed–Solomon codes are a group of

error-correcting code In computing, telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable or noisy communication channels. The centra ...

s that were introduced by Irving S. Reed and Gustave Solomon in 1960. They have many applications, including consumer technologies such as

MiniDisc MiniDisc (MD) is an erasable magneto-optical disc-based data storage format offering a capacity of 60, 74, or 80 minutes of digitized audio. Sony announced the MiniDisc in September 1992 and released it in November of that year for sale i ...

s, CDs,

DVD The DVD (common abbreviation for digital video disc or digital versatile disc) is a digital optical disc data storage format. It was invented and developed in 1995 and first released on November 1, 1996, in Japan. The medium can store any ki ...

Blu-ray Blu-ray (Blu-ray Disc or BD) is a digital optical disc data storage format designed to supersede the DVD format. It was invented and developed in 2005 and released worldwide on June 20, 2006, capable of storing several hours of high-defin ...

discs, QR codes,

Data Matrix A Data Matrix is a two-dimensional code consisting of black and white "cells" or dots arranged in either a square or rectangular pattern, also known as a matrix. The information to be encoded can be text or numeric data. Usual data size is fro ...

data transmission Data communication, including data transmission and data reception, is the transfer of data, signal transmission, transmitted and received over a Point-to-point (telecommunications), point-to-point or point-to-multipoint communication chann ...

technologies such as

DSL Digital subscriber line (DSL; originally digital subscriber loop) is a family of technologies that are used to transmit digital data over telephone lines. In telecommunications marketing, the term DSL is widely understood to mean asymmetric di ...

and

WiMAX Worldwide Interoperability for Microwave Access (WiMAX) is a family of wireless broadband communication standards based on the IEEE 802.16 set of standards, which provide physical layer (PHY) and media access control (MAC) options. The WiMA ...

broadcast Broadcasting is the data distribution, distribution of sound, audio audiovisual content to dispersed audiences via a electronic medium (communication), mass communications medium, typically one using the electromagnetic spectrum (radio waves), ...

systems such as satellite communications, DVB and ATSC, and storage systems such as

RAID 6 In computer storage, the standard RAID levels comprise a basic set of RAID ("redundant array of independent disks" or "redundant array of inexpensive disks") configurations that employ the techniques of striping, mirroring, or parity to create la ...

. 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 (but not correct) any combination of up to erroneous symbols, ''or'' locate and correct up to erroneous symbols at unknown locations. As an

erasure code In coding theory, an erasure code is a forward error correction (FEC) code under the assumption of bit erasures (rather than bit errors), which transforms a message of ''k'' symbols into a longer message (code word) with ''n'' symbols such that ...

, 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 Burst may refer to: *Burst mode (disambiguation), a mode of operation where events occur in rapid succession **Burst transmission, a term in telecommunications **Burst switching, a feature of some packet-switched networks **Bursting, a signaling mo ...

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 The MIT Lincoln Laboratory, located in Lexington, Massachusetts, is a United States Department of Defense federally funded research and development center chartered to apply advanced technology to problems of national security. Research and dev ...

. 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 (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 Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician best remembered for his contribution to the classification of finite simple groups. Gorenstein mastered calculus at age 12 and subsequently matriculated at ...

and Neal Zierler was described in an

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 (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 code In coding theory, a cyclic code is a block code, where the circular shifts of each codeword gives another word that belongs to the code. They are error-correcting codes that have algebraic properties that are convenient for efficient error detecti ...

s. In 1969, an improved BCH scheme decoder was developed by

Elwyn Berlekamp Elwyn Ralph Berlekamp (September 6, 1940 – April 9, 2019) was a professor of mathematics and computer science at the University of California, Berkeley.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 arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of Bézout's id ...

In 1977, Reed–Solomon codes were implemented in the

Voyager program The Voyager program is an American scientific program that employs two interstellar probes, ''Voyager 1'' and ''Voyager 2''. They were launched in 1977 to take advantage of a favorable planetary alignment to explore the two gas giants Jupiter ...

in the form of concatenated error correction codes. The first commercial application in mass-produced consumer products appeared in 1982 with the

compact disc The compact disc (CD) is a Digital media, digital optical disc data storage format co-developed by Philips and Sony to store and play digital audio recordings. It employs the Compact Disc Digital Audio (CD-DA) standard and was capable of hol ...

, where two interleaved Reed–Solomon codes are used. Today, Reed–Solomon codes are widely implemented in

digital storage Data storage is the recording (storing) of information (data) in a storage medium. Handwriting, Phonograph record, phonographic recording, magnetic tape, and optical discs are all examples of storage media. Biological molecules such as RNA ...

devices and

digital communication Data communication, including data transmission and data reception, is the transfer of data, transmitted and received over a point-to-point or point-to-multipoint communication channel. Examples of such channels are copper wires, optical ...

standards, though they are being slowly replaced by Bose–Chaudhuri–Hocquenghem (BCH) codes. For example, Reed–Solomon codes are used in the

Digital Video Broadcasting Digital Video Broadcasting (DVB) is a set of international open standards for digital television. DVB standards are maintained by the DVB Project, an international industry consortium, and are published by a Joint Technical Committee (JTC) o ...

(DVB) standard

DVB-S Digital Video Broadcasting – Satellite (DVB-S) is the original DVB standard for satellite television and dates from 1995, in its first release, while development lasted from 1993 to 1997. The first commercial applications were by Canal+ in ...

, in conjunction with a convolutional inner code, but BCH codes are used with

LDPC Low-density parity-check (LDPC) codes are a class of error correction codes which (together with the closely-related turbo codes) have gained prominence in coding theory and information theory since the late 1990s. The codes today are widely ...

in its successor,

DVB-S2 Digital Video Broadcasting - Satellite - Second Generation (DVB-S2) is a digital television broadcast standard that has been designed as a successor for the popular DVB-S system. It was developed in 2003 by the Digital Video Broadcasting Proj ...

. In 1986, an original scheme decoder known as the

Berlekamp–Welch algorithm The Berlekamp–Welch algorithm, also known as the Welch–Berlekamp algorithm, is named for Elwyn R. Berlekamp and Lloyd R. Welch. This is a decoder algorithm that efficiently corrects errors in Reed–Solomon error correction, Reed–Solomon code ...

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 coding theory, list decoding is an alternative to unique decoding of error-correcting codes in the presence of many errors. If a code has relative distance \delta, then it is possible in principle to recover an encoded message when up to \de ...

). In 2002, another original scheme decoder was developed by Shuhong Gao, based on the

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

. It was the first use of strong error correction coding in a mass-produced consumer product, and DAT and

use similar schemes. In the CD, two layers of Reed–Solomon coding separated by a 28-way

convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...

al interleaver yields a scheme called Cross-Interleaved Reed–Solomon Coding (

CIRC Circ or CIRC may refer to: * Čirč, a village and municipality in northern Slovakia * Circ (duo), an American music duo * Cook Islands Round Cup, top division association football league in the Cook Islands Commercial * China Insurance Regulato ...

). 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 Parchive (a portmanteau of parity archive, and formally known as Parity Volume Set Specification) is an erasure code system that produces par files for checksum verification of data integrity, with the capability to perform data recovery operatio ...

files which are commonly posted accompanying multimedia files on

USENET Usenet (), a portmanteau of User's Network, is a worldwide distributed discussion system available on computers. It was developed from the general-purpose UUCP, Unix-to-Unix Copy (UUCP) dial-up network architecture. Tom Truscott and Jim Elli ...

. The distributed online storage service Wuala (discontinued in 2015) 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 Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...

-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 2''K'', 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 In computing, telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable or noisy communication channels. The centra ...

Space transmission

One significant application of Reed–Solomon coding was to encode the digital pictures sent back by the

. Voyager introduced Reed–Solomon coding concatenated with

convolutional code In telecommunication, a convolutional code is a type of error-correcting code that generates parity symbols via the sliding application of a boolean polynomial function to a data stream. The sliding application represents the 'convolution' of th ...

s, 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 ''Mars Pathfinder'' was an American robotic spacecraft that landed a base station with a rover (space exploration), roving probe on Mars in 1997. It consisted of a Lander (spacecraft), lander, renamed the Carl Sagan Memorial Station, and a ligh ...

Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a poly ...

Mars Exploration Rover NASA's Mars Exploration Rover (MER) mission was a robotic space mission involving two Mars rovers, ''Spirit (rover), Spirit'' and ''Opportunity (rover), Opportunity'', exploring the planet Mars. It began in 2003 with the launch of the two rove ...

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:

Constructions (encoding)

The Reed–Solomon code is actually a family of codes, where every code is characterised by three parameters: an

alphabet An alphabet is a standard set of letter (alphabet), letters written to represent particular sounds in a spoken language. Specifically, letters largely correspond to phonemes as the smallest sound segments that can distinguish one word from a ...

size , a block length , and a message length '','' with

k < n \leq q

. The set of alphabet symbols is interpreted as the

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

F

of order

q

, and thus,

q

must be a

prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 1 ...

. 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 = \frac

is some constant, and furthermore, the block length is either equal to the alphabet size or one less than it, i.e.,

n=q

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 and Solomon, 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 symbols (each within the q-sized alphabet) are treated as the coefficients 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 \leq 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

\

\

, or for

n < q

\

, ... , where

\alpha

is a primitive element of

F

. Formally, the set

\mathbf

of codewords of the Reed–Solomon code is defined as follows:

\mathbf
 = \Bigl\\,.

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-1

points but are not equal, and thus, the

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...

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 In coding theory, the Singleton bound, named after the American mathematician Richard Collom Singleton (1928–2007), is a relatively crude upper bound on the size of an arbitrary block code C with block length n, size M and minimum distance d. It ...

, ''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 code In coding theory, the Singleton bound, named after the American mathematician Richard Collom Singleton (1928–2007), is a relatively crude upper bound on the size of an arbitrary block code C with block length n, size M and minimum distance d. It ...

s. 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 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 In coding theory, a systematic code is any error-correcting code in which the input data are embedded in the encoded output. Conversely, in a non-systematic code the output does not contain the input symbols. Systematic codes have the advantage th ...

, 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 and Solomon, the message

m=(m_0,\dots,m_) \in F^k

is mapped to the polynomial

p_m

with

p_m(a) = \sum_^ m_i a^ \,.

The codeword of

m

is obtained by evaluating

p_m

n

different points

a_0, \dots, a_

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(m) = \begin
p_m(a_0) \\
p_m(a_1) \\
\cdots \\
p_m(a_)
\end

This function

C

is a

linear mapping In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vec ...

, that is, it satisfies

C(m) = Am

for the following

n \times k

-matrix

A

with elements from

F

C(m) = Am = \begin
1 & a_0 & a_0^2 & \dots & a_0^ \\
1 & a_1 & a_1^2 & \dots & a_1^ \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & a_ & a_^2 & \dots  & a_^
\end
\begin
m_0 \\
m_1 \\
\vdots \\
m_
\end

This matrix is a

Vandermonde matrix In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an (m + 1) \times (n + 1) matrix :V = V(x_0, x_1, \cdots, x_m) = \begin 1 & x_0 & x_0^2 & \dot ...

over

F

. In other words, the Reed–Solomon code is a

linear code In coding theory, a linear code is an error-correcting code for which any linear combination of Code word (communication), codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although t ...

, and in the classical encoding procedure, its

generator matrix In coding theory, a generator matrix is a matrix whose rows form a basis for a linear code. The codewords are all of the linear combinations of the rows of this matrix, that is, the linear code is the row space of its generator matrix. Terminolo ...

A

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

There are alternative encoding procedures that produce a

systematic Systematic may refer to: Science * Short for systematic error * Systematic fault * Systematic bias, errors that are introduced by an inaccuracy inherent to the system Economy * Systematic trading, a way of defining trade goals, risk control ...

Reed–Solomon code. One method uses

Lagrange interpolation In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' ...

to compute polynomial

p_m

such that

p_m(a_i) = m_i \text i\in\.

Then

p_m

is evaluated at the other points

a_k, \dots, a_

C(m) = \begin
p_m(a_0) \\
p_m(a_1) \\
\cdots \\
p_m(a_)
\end

This function

C

is a linear mapping. To generate the corresponding systematic encoding matrix G, multiply the Vandermonde matrix A by the inverse of A's left square submatrix.

G =
(A\text)^ \cdot A
=
\begin
1         & 0      & 0         & \dots  & 0      & g_ & \dots  & g_ \\
0         & 1      & 0         & \dots  & 0      & g_ & \dots  & g_ \\
0         & 0      & 1         & \dots  & 0      & g_ & \dots  & g_ \\
\vdots    & \vdots & \vdots    &        & \vdots & \vdots    &        & \vdots  \\
0         & \dots  & 0         & \dots  & 1      & g_ & \dots  & g_
\end

C(m) = Gm

for the following

n \times k

-matrix

G

with elements from

F

C(m) = Gm =
\begin
1         & 0      & 0         & \dots  & 0      & g_ & \dots  & g_ \\
0         & 1      & 0         & \dots  & 0      & g_ & \dots  & g_ \\
0         & 0      & 1         & \dots  & 0      & g_ & \dots  & g_ \\
\vdots    & \vdots & \vdots    &        & \vdots & \vdots    &        & \vdots  \\
0         & \dots  & 0         & \dots  & 1      & g_ & \dots  & g_
\end\begin
m_0 \\
m_1 \\
\vdots \\
m_
\end

Discrete Fourier transform and its inverse

discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...

is essentially the same as the encoding procedure; it uses the generator polynomial

p_m

to map a set of evaluation points into the message values as shown above:

C(m) = \begin
p_m(a_0) \\
p_m(a_1) \\
\cdots \\
p_m(a_)
\end

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

a_0, \dots, a_ = \

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 message is interpreted as the coefficients of a polynomial

p(x)

. The sender computes a related polynomial

s(x)

of degree

n-1

where

n \le q-1

and sends the polynomial

s(x)

. The polynomial

s(x)

is constructed by multiplying the message polynomial

p(x)

, which has degree

k-1

, with a

generator polynomial In coding theory, a polynomial code is a type of linear code whose set of valid code words consists of those polynomials (usually of some fixed length) that are divisible by a given fixed polynomial (of shorter length, called the ''generator polyno ...

g(x)

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 sequential powers of the Galois field primitive

\alpha

g(x) = \left(x-\alpha^\right) \left(x-\alpha^\right) \cdots \left(x-\alpha^\right) = g_0 + g_1x + \cdots + g_x^ + x^

For a "narrow sense code",

i = 1

\mathbf =
\left\ .

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, and simply appends error correcting symbols as a suffix. 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)

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).

The remainder has degree at most

t-1

, whereas the coefficients of

x^,x^,\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

, that is, they are divisible by the generator polynomial

g(x)

s(x) \equiv p(x) \cdot x^t - s_r(x) \equiv s_r(x) - s_r(x) \equiv 0 \mod g(x)\,.

This function

s

is a linear mapping. To generate the corresponding systematic encoding matrix G, set G's left square submatrix to the identity matrix and then encode each row:

G =
\begin
1         & 0      & 0         & \dots  & 0      & g_ & \dots  & g_ \\
0         & 1      & 0         & \dots  & 0      & g_ & \dots  & g_ \\
0         & 0      & 1         & \dots  & 0      & g_ & \dots  & g_ \\
\vdots    & \vdots & \vdots    &        & \vdots & \vdots    &        & \vdots  \\
0         & \dots  & 0         & \dots  & 1      & g_ & \dots  & g_
\end

Ignoring leading zeroes, the last row =

g(x)

C(m) = mG

for the following

n \times k

-matrix

G

with elements from

F

C(m) = mG = \begin m_0 & m_1 & \ldots & m_\end
\begin
1         & 0      & 0         & \dots  & 0      & g_ & \dots  & g_ \\
0         & 1      & 0         & \dots  & 0      & g_ & \dots  & g_ \\
0         & 0      & 1         & \dots  & 0      & g_ & \dots  & g_ \\
\vdots    & \vdots & \vdots    &        & \vdots & \vdots    &        & \vdots  \\
0         & \dots  & 0         & \dots  & 1      & g_ & \dots  & g_
\end

Properties

The Reed–Solomon code is a 'n'', ''k'', ''n'' − ''k'' + 1code; in other words, it is a

linear block code In coding theory, block codes are a large and important family of error-correcting codes that encode data in blocks. There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract defi ...

of length ''n'' (over ''F'') with

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

''k'' and minimum

Hamming distance In information theory, the Hamming distance between two String (computer science), strings or vectors of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number ...

d_ = 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

. 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 Demodulation is the process of extracting the original information-bearing signal from a carrier wave. A demodulator is an electronic circuit (or computer program in a software-defined radio) that is used to recover the information content from ...

signal-to-noise ratio Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to noise power, often expressed in deci ...

s)—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 is satisfied, where

E

is the number of errors and

S

is the number of erasures in the block. RS BER

The theoretical error bound can be described via the following formula for the AWGN channel for FSK:

P_b \approx \frac\frac\sum_^n \ell P_s^\ell(1-P_s)^

and for other modulation schemes:

P_b \approx \frac\frac\sum_^n \ell  P_s^\ell(1-P_s)^

where

t = \frac(d_-1)

P_s = 1-(1-s)^h

h = \frac

s

is the symbol error rate in uncoded AWGN case and

M

is the modulation order. 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

s. 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

, 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 (''see 'Remarks' at the end of this section''). 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\

, where

q=, F,

. Each polynomial

p

over

F

gives rise to a codeword

(p(\alpha^1),\dots,p(\alpha^))

. 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^))

; since

\alpha^=\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 QR code, Ver 3 (29×29) uses interleaved blocks. The message has 26 data bytes and is encoded using two Reed-Solomon code blocks. Each block is a (255,233) Reed Solomon code shortened to a (35,13) code. 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.

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. The Gorenstein–Zierler decoder and the related work on BCH codes are described in the book ''Error Correcting Codes'' by W. Wesley Peterson (1961).

Formulation

The transmitted message,

(c_0, \ldots, c_i, \ldots,c_)

, is viewed as the coefficients of a polynomial

s(x) = \sum_^ c_i x^i.

As a result of the Reed–Solomon encoding procedure, ''s''(''x'') is divisible by the generator polynomial

g(x) = \prod_^ (x - \alpha^j),

where ''α'' is a primitive element. Since ''s''(''x'') is a multiple of the generator ''g''(''x''), it follows that it "inherits" all its roots:

s(x) \bmod (x - \alpha^j) = g(x) \bmod (x - \alpha^j) = 0.

Therefore,

s(\alpha^j) = 0,\ j = 1, 2, \ldots, n - k.

The transmitted polynomial is corrupted in transit by an error polynomial

e(x) = \sum_^ e_i x^i

to produce the received polynomial

r(x) = s(x) + e(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_^\nu e_ x^.

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 originally sent message ''s''(''x'').

Syndrome decoding

The decoder starts by evaluating the polynomial as received at points

\alpha^1 \dots \alpha^

. We call the results of that evaluation the "syndromes" ''S''_''j''. They are defined as

\begin
  S_j &= r(\alpha^j) = s(\alpha^j) + e(\alpha^j) = 0 + e(\alpha^j) \\
      &= e(\alpha^j) \\
      &= \sum_^\nu e_ ^, \quad j = 1, 2, \ldots, n - k.
 \end

Note that

s(\alpha^j) = 0

because

s(x)

has roots at

\alpha^j

, as shown in the previous section. 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.

Error locators and error values

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

X_k = \alpha^, \quad Y_k = e_.

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

S_j = \sum_^\nu Y_k X_k^j.

This definition of the syndrome values is equivalent to the previous since

^ = \alpha^ = ^j = X_k^j

. The syndromes give a system of equations in 2''ν'' 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

\begin
X_1^1 & X_2^1 & \cdots & X_\nu^1 \\
X_1^2 & X_2^2 & \cdots & X_\nu^2 \\
\vdots & \vdots & \ddots & \vdots \\
X_1^ & X_2^ & \cdots & X_\nu^ \\
\end
\begin
Y_1 \\ Y_2 \\ \vdots \\ Y_\nu
\end
=
\begin
S_1 \\ S_2 \\ \vdots \\ S_
\end,

which can easily be solved for the ''Y_k'' error values. 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'' In the variant of this algorithm where the locations of the errors are already known (when it is being used as an

), this is the end. The error locations (''X_k'') are already known by some other method (for example, in an FM transmission, the sections where the bitstream was unclear or overcome with interference are probabilistically determinable from frequency analysis). In this scenario, up to

n - k

errors can be corrected. The rest of the algorithm serves to locate the errors and will require syndrome values up to

2\nu

, instead of just the

\nu

used thus far. This is why twice as many error-correcting symbols need to be added as can be corrected without knowing their locations.

Error locator polynomial

There is a linear recurrence relation that gives rise to a system of linear equations. Solving those equations identifies those error locations ''X_k''. Define the error locator polynomial as

\Lambda(x) = \prod_^\nu (1 - x X_k ) = 1 + \Lambda_1 x^1 + \Lambda_2 x^2 + \cdots + \Lambda_\nu x^\nu.

The zeros of are the reciprocals

X_k^

. This follows from the above product notation construction, since if

x = X_k^

, then one of the multiplied terms will be zero,

(1 - X_k^ \cdot X_k) = 1 - 1 = 0

, making the whole polynomial evaluate to zero:

\Lambda(X_k^) = 0.

Let

j

be any integer such that

1 \leq j \leq \nu

. Multiply both sides by

Y_k X_k^

, and it will still be zero:

\begin
 & Y_k X_k^ \Lambda(X_k^) = 0, \\
 & Y_k X_k^ (1 + \Lambda_1 X_k^ + \Lambda_2 X_k^ + \cdots + \Lambda_\nu X_k^) = 0, \\
 & Y_k X_k^ + \Lambda_1 Y_k X_k^ X_k^ + \Lambda_2 Y_k X_k^ X_k^ + \cdots + \Lambda_\nu Y_k X_k^ X_k^ = 0, \\
 & Y_k X_k^ + \Lambda_1 Y_k X_k^ + \Lambda_2 Y_k X_k^ + \cdots + \Lambda_\nu Y_k X_k^j = 0.
\end

Sum for ''k'' = 1 to ''ν'', and it will still be zero:

\sum_^\nu (Y_k X_k^ + \Lambda_1 Y_k X_k^ + \Lambda_2 Y_k X_k^ + \cdots + \Lambda_ Y_k X_k^j) = 0.

Collect each term into its own sum:

\left(\sum_^\nu Y_k X_k^\right) + \left(\sum_^\nu \Lambda_1 Y_k X_k^\right) + \left(\sum_^\nu \Lambda_2 Y_k X_k^\right) + \cdots + \left(\sum_^\nu \Lambda_\nu Y_k X_k^j\right) = 0.

Extract the constant values of

\Lambda

that are unaffected by the summation:

\left(\sum_^\nu Y_k X_k^\right) + \Lambda_1 \left(\sum_^\nu Y_k X_k^\right) + \Lambda_2 \left(\sum_^\nu Y_k X_k^\right) + \cdots + \Lambda_\nu \left(\sum_^\nu Y_k X_k^j\right) = 0.

These summations are now equivalent to the syndrome values, which we know and can substitute in. This therefore reduces to

S_ + \Lambda_1 S_ + \cdots + \Lambda_ S_ + \Lambda_\nu S_j = 0.

Subtracting

S_

from both sides yields

S_j \Lambda_\nu + S_ \Lambda_ + \cdots + S_ \Lambda_1 = -S_.

Recall that ''j'' was chosen to be any integer between 1 and ''v'' inclusive, and this equivalence is true for all such values. Therefore, we have ''v'' linear equations, not just one. This system of linear equations can therefore be solved for the coefficients Λ_''i'' of the error-location polynomial:

\begin
  S_1 & S_2 & \cdots & S_\nu \\
  S_2 & S_3 & \cdots & S_ \\
  \vdots & \vdots & \ddots & \vdots \\
  S_\nu & S_ & \cdots & S_
 \end
 \begin
  \Lambda_\nu \\ \Lambda_ \\ \vdots \\ \Lambda_1
 \end
 =
 \begin
  -S_ \\ -S_ \\ \vdots \\ -S_
 \end.

The above assumes that 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.

Find the roots of 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 ''X_k'' are the reciprocals of those roots. The order of coefficients of the error location polynomial can be reversed, in which case the roots of that reversed polynomial are the error locators

X_k

(not their reciprocals

X_k^

). Chien search is an efficient implementation of this step.

Calculate the error values

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

Calculate the error locations

Calculate ''i_k'' by taking the log base

\alpha

of ''X_k''. This is generally done using a precomputed lookup table.

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 originally sent message ''s''(''x''), with errors corrected.

Example

Consider the Reed–Solomon code defined in with and (this is used in

PDF417 PDF417 is a stacked linear barcode format used in a variety of applications such as transport, identification cards, and inventory management. "PDF" stands for ''Portable Data File'', while "417" signifies that each pattern in the code consis ...

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 + 809 x^3 + 723 x^2 + 568 x + 522.

If the message polynomial is , then a systematic codeword is encoded as follows:

s_r(x) = p(x) \, x^t \bmod g(x) = 547 x^3 + 738 x^2 + 442 x + 455,

s(x) = p(x) \, x^t - s_r(x) = 3 x^6 + 2 x^5 + 1 x^4 + 382 x^3 + 191 x^2 + 487 x + 474.

Errors in transmission might cause this to be received instead:

r(x) = s(x) + e(x) = 3 x^6 + 2 x^5 + 123 x^4 + 456 x^3 + 191 x^2 + 487 x + 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,\quad S_3 = r(3^3) = 762,\quad S_4 = r(3^4) = 925,

yielding the system

\begin
  732 & 637 \\
  637 & 762
 \end
 \begin
  \Lambda_2 \\ \Lambda_1
 \end
 =
 \begin
  -762 \\ -925
 \end
 =
 \begin
  167 \\ 004
 \end.

Using

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can a ...

\begin
  001 & 000 \\
  000 & 001
 \end
 \begin
  \Lambda_2 \\ \Lambda_1
 \end
 =
 \begin
  329 \\ 821
 \end,

\Lambda(x) = 329 x^2 + 821 x + 001,

with roots ''x''₁ = 757 = 3⁻³ and ''x''₂ = 562 = 3⁻⁴. The coefficients can be reversed:

R(x) = 001 x^2 + 821 x + 329,

to produce roots 27 = 3³ and 81 = 3⁴ with positive exponents, but typically this isn't used. The logarithm of the inverted roots corresponds 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:

\Omega(x) = S(x) \Lambda(x) \bmod x^4 = 546 x + 732,

\Lambda'(x) = 658 x + 821,

e_1 = -\Omega(x_1)/\Lambda'(x_1) = 074,

e_2 = -\Omega(x_2)/\Lambda'(x_2) = 122.

Subtracting

e_1 x^3 + e_2 x^4 = 74x^3 + 122x^4

from the received polynomial ''r''(''x'') 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_ + \Lambda_1 \  S_ + \cdots + \Lambda_e \  S_

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

Sugiyama decoder

Another iterative method for calculating both the error locator polynomial and the error value polynomial is based on Sugiyama's adaptation of the

. Define ''S''(''x''), Λ(''x''), and Ω(''x'') for ''t'' syndromes and ''e'' errors:

\Omega(x) &= \Omega_ x^ + \Omega_ x^ + \cdots + \Omega_ x + \Omega_ \end

The key equation is:

\Lambda(x) S(x) = Q(x) x^ + \Omega(x)

For ''t'' = 6 and ''e'' = 3:

\begin
\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
=
\begin
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

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 ''R''_''i''(''x'') < ''t''/2, then ''B''(''x'') and ''Q''(''x'') don't need to be saved, so the algorithm becomes: ''R''₋₁ := ''x''^''t'' ''R''₀ := ''S''(''x'') ''A''₋₁ := 0 ''A''₀ := 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): ''A''_''i''(0) is the constant (low order) term of A_i.

Example

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

Decoder using discrete Fourier transform

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:

R_j = E_j = S_j = r(\alpha^j) \qquad \text 1 \le j \le 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

E_t

, the error locator polynomial, and these formulas

\begin
E_0 &= - \frac(E_ + \Lambda_1 E_ + \cdots + \Lambda_ E_) \\
E_j &= -(\Lambda_1 E_ + \Lambda_2 E_ + \cdots + \Lambda_v E_)
& \text t < j < n
\end

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

states that the minimum distance of a linear block code of size (,) is upper-bounded by . The distance was usually understood to limit the error-correction capability to . The Reed–Solomon code achieves this bound with equality, and can thus correct up to errors. However, this error-correction bound is not exact. In 1999,

Madhu Sudan Madhu Sudan (born 12 September 1966) is an Indian-American computer scientist. He has been a Gordon McKay Professor of Computer Science at the Harvard John A. Paulson School of Engineering and Applied Sciences since 2015. Career He received hi ...

and

Venkatesan Guruswami Venkatesan Guruswami (born 1976) is a senior scientist at the Simons Institute for the Theory of Computing and Professor of EECS and Mathematics at the University of California, Berkeley. He did his high schooling at Padma Seshadri Bala Bhav ...

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. 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 In coding theory, list decoding is an alternative to unique decoding of error-correcting codes for large error rates. The notion was proposed by Elias in the 1950s. The main idea behind list decoding is that the decoding algorithm instead of out ...

algorithm) and is based on interpolation and factorization of polynomials over and its extensions. In 2023, building on three exciting works, coding theorists showed that Reed-Solomon codes defined over random evaluation points can actually achieve

list decoding In coding theory, list decoding is an alternative to unique decoding of error-correcting codes for large error rates. The notion was proposed by Elias in the 1950s. The main idea behind list decoding is that the decoding algorithm instead of outp ...

capacity (up to errors) over linear size alphabets with high probability. However, this result is combinatorial rather than algorithmic.

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

's confidence in the correctness of the symbol. The advent of

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 Alexander Vardy (, ; 12 November 1963 - 11 March 2022) was a Russian-born and Israeli-educated electrical engineer known for his expertise in coding theory.. He held the Jack Keil Wolf Endowed Chair in Electrical Engineering at the University of Ca ...

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.

MATLAB example

Encoder

Here we present a simple

MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...

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( sg 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, alpha .^ (k); end end

Decoder

Now the decoding part: function ecoded, error_pos, error_mag, g, S= rsDecoder(encoded, m, prim_poly, n, k) % RSDECODER Decode a Reed-Solomon encoded message % Example: % ec, ~, ~, ~, ~= 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( eros(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 uotient, 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 = eros(1, k - len) x end end

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 and Solomon described a theoretical decoder that corrected errors by finding the most popular message polynomial. The decoder only knows the set of values

a_1

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 In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

\binom =

, and the number of subsets is infeasible for even modest codes. For a code that can correct 3 errors, the naïve theoretical decoder would examine 359 billion subsets.

Berlekamp Welch decoder

In 1986, a decoder known as the

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 , where 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 : If the message polynomial is : The codeword is : Errors in transmission might cause this to be received instead. : The key equation is: : Assume maximum number of errors: . The key equation becomes: :

\begin
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
\begin
e_0 \\
e_1 \\
q_0 \\
q_1 \\
q_2 \\
q_3 \\
q_4
\end
=
\begin
000 \\
923 \\
437 \\
541 \\
017 \\
637 \\
289
\end

Using

\begin
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
\begin
e_0 \\
e_1 \\
q_0 \\
q_1 \\
q_2 \\
q_3 \\
q_4
\end
=
\begin
006 \\
924 \\
006 \\
007 \\
009 \\
916 \\
003
\end

: : : Recalculate where to correct resulting in the corrected codeword: :

Gao decoder

In 2002, an improved decoder was developed by Shuhong Gao, based on the extended Euclid algorithm. https://www.math.clemson.edu/~sgao/papers/RS.pdf

Example

R_ = \prod_^n (x - a_i)

R_0 =

Lagrange interpolation of

(a_i, b(a_i))

for

i = 1

n

A_ = 0

A_ = 1

*generate

R_

and

A_

until degree of

R_ < (n+k)/2

, for this example

(n+k)/2 = (7+3)/2 = 5

: : To duplicate the polynomials generated by Berlekamp Welsh, divide ''Q''(''x'') and ''E''(''x'') by most significant coefficient of ''E''(''x'') = 708. : : : Recalculate where to correct resulting in the corrected codeword: :

Notes

References

External links

Information and tutorials

* *
Introduction to Reed–Solomon codes: principles, architecture and implementation
(CMU) * *

Implementations

FEC library in C by Phil Karn (aka KA9Q) includes Reed–Solomon codec, both arbitrary and optimized (223,255) version

Open Source C++ Reed–Solomon Soft Decoding library

Matlab implementation of errors and-erasures Reed–Solomon decoding
{{DEFAULTSORT:Reed-Solomon error correction Error detection and correction Coding theory

History

Applications

Data storage

Bar code

Data transmission

Space transmission

Constructions (encoding)

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

BCH view decoders

Peterson–Gorenstein–Zierler decoder

Formulation

Syndrome decoding

Error locators and error values

Error locator polynomial

Find the roots of the error locator polynomial

Calculate the error values

Calculate the error locations

Fix the errors

Example

Berlekamp–Massey decoder

Example

Sugiyama decoder

Example

Decoder using discrete Fourier transform

Decoding beyond the error-correction bound

Soft-decoding

MATLAB example

Encoder

Decoder

Reed Solomon original view decoders

Theoretical decoder

Berlekamp Welch decoder

Example

Gao decoder

Example

See also

Notes

References

External links

Information and tutorials

Implementations