Algorithm Error Correction
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BCH 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 Decoding Berlekamp–Massey Euclidean ecc algorithm et al. Properties Maximum-distance separable code v t e Reed–Solomon codes are a group
Reed Solomon Error Correction Algorithm
of error-correcting codes that were introduced by Irving S. Reed and Gustave Solomon in 1960.[1] They have many applications, the most prominent
Error Detection And Correction Algorithms
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
Hamming Distance Error Correction
also used in satellite communication. In coding theory, the Reed–Solomon code belongs to the class of non-binary cyclic error-correcting codes. The Reed–Solomon code is based on univariate polynomials over finite fields. It is 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 error correction and detection can correct up to t known erasures, or it can detect and correct combinations of errors and erasures. Furthermore, Reed–Solomon codes are 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. Contents 1 History 2 Applications 2.1 Data storage 2.2 Bar code 2.3 Data transmission 2.4 Space transmission 3 Constructions 3.1 Reed & Solomon's original view: The codeword as a sequence of values 3.1.1 Simple encoding procedure: The message as a sequence of coefficients 3.1.2 Systematic encoding procedure: The message as an initial sequence of values 3.1.3 Theoretical decoding procedure 3.2 The BCH view: The codeword as a sequence of coefficients 3.2.1 Systematic encoding procedure 3.3 Duality of the two views - discrete Fourier transform 3.4 Remarks 4 Properties 5 Error correction algorithms 5.1 Peterson–Gorenstein–Zierler decoder 5.1.1 Syndrome decoding 5.1.2 Error locators and error values 5.1.3 Error locator polynomial 5.1.4 Obtain the error locators from the error locator polynomial 5.1.5 Calculate the error locations 5.1.6 Calculate the error values 5.1.7 Fix the errors 5.2 Berlekamp–Massey decoder 5.2.1 Example 5.3 Euclidean decoder 5.3.1 Example 5.4 Decoder using discrete Fourier transform 5.5 Decoding beyond the error-corr
(Discuss) Proposed since January 2015. In telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding[1] is a technique used for controlling errors in data transmission over unreliable or noisy error correction code communication channels. The central idea is the sender encodes the message in a error correction techniques redundant way by using an error-correcting code (ECC). The American mathematician Richard Hamming pioneered this field in the 1940s and reed solomon error correction invented the first error-correcting code in 1950: the Hamming (7,4) code.[2] The redundancy allows the receiver to detect a limited number of errors that may occur anywhere in the message, and often to https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction correct these errors without retransmission. FEC gives the receiver the ability to correct errors without needing a reverse channel to request retransmission of data, but at the cost of a fixed, higher forward channel bandwidth. FEC is therefore applied in situations where retransmissions are costly or impossible, such as one-way communication links and when transmitting to multiple receivers in multicast. FEC information is usually added to https://en.wikipedia.org/wiki/Forward_error_correction mass storage devices to enable recovery of corrupted data, and is widely used in modems. FEC processing in a receiver may be applied to a digital bit stream or in the demodulation of a digitally modulated carrier. For the latter, FEC is an integral part of the initial analog-to-digital conversion in the receiver. The Viterbi decoder implements a soft-decision algorithm to demodulate digital data from an analog signal corrupted by noise. Many FEC coders can also generate a bit-error rate (BER) signal which can be used as feedback to fine-tune the analog receiving electronics. The noisy-channel coding theorem establishes bounds on the theoretical maximum information transfer rate of a channel with some given noise level. Some advanced FEC systems come very close to the theoretical maximum. The maximum fractions of errors or of missing bits that can be corrected is determined by the design of the FEC code, so different forward error correcting codes are suitable for different conditions. Contents 1 How it works 2 Averaging noise to reduce errors 3 Types of FEC 4 Concatenated FEC codes for improved performance 5 Low-density parity-check (LDPC) 6 Turbo codes 7 Local decoding and testing of codes 8 Interleav
Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Tue Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch http://mathworld.wolfram.com/Error-CorrectingCode.html Discrete Mathematics>Coding Theory> Interactive Entries>Interactive Demonstrations> Error-Correcting Code An error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced can be detected and corrected (within certain limitations) based on the remaining numbers. The study of error-correcting codes and the associated mathematics is known as coding theory. Error detection is much simpler than error correction, and error correction one or more "check" digits are commonly embedded in credit card numbers in order to detect mistakes. Early space probes like Mariner used a type of error-correcting code called a block code, and more recent space probes use convolution codes. Error-correcting codes are also used in CD players, high speed modems, and cellular phones. Modems use error detection when they compute checksums, which reed solomon error are sums of the digits in a given transmission modulo some number. The ISBN used to identify books also incorporates a check digit. A powerful check for 13 digit numbers consists of the following. Write the number as a string of digits . Take and double. Now add the number of digits in odd positions which are to this number. Now add . The check number is then the number required to bring the last digit to 0. This scheme detects all single digit errors and all transpositions of adjacent digits except 0 and 9. Let denote the maximal number of (0,1)-vectors having the property that any two of the set differ in at least places. The corresponding vectors can correct errors. is the number of s with precisely 1s (Sloane and Plouffe 1995). Since it is not possible for -vectors to differ in places and since -vectors which differ in all places partition into disparate sets of two, (1) Values of can be found by labeling the (0,1)--vectors, finding all unordered pairs of -vectors which differ from each other in at least places, forming a g