|
| |
| | Error correction and detection - Wikipedia, the free encyclopedia |
 | | Information theory tells us that whatever be the probability of error in transmission or storage, it is possible to construct error-correcting codes in which the likelihood of failure is arbitrarily low, although this requires adding increasing amounts of redundant data to the original, which might not be practical when the error probability is very high. | | | In digital telecommunications, channel coding is a pre-transmission mapping applied to a digital signal or data file, usually with an error-detection or an error-correction code. | | | Say now an error occurs in the first byte, that "001011010" is received as "101011010". |
|
http://en.wikipedia.org/wiki/Error_correcting_code
|
|
| |
| | NodeWorks - Encyclopedia: Error-correcting code |
| | In information theory and coding, an error-correcting code or ECC is a code in which each data signal conforms to specific rules of construction so that departures from this construction in the received signal can generally be automatically detected and corrected. | | | Shannon's theorem is an important theory in error correction which describes the maximum attainable efficiency of an error-correcting scheme versus the levels of noise interference expected. | | | Note 1: If the number of errors is less than or equal to the maximum correctable threshold of the code, all errors will be corrected. |
|
http://pedia.nodeworks.com/E/ER/ERR/Error-correcting_code
|
|
| |
| | The Error Correcting Codes (ECC) Page |
| | This BCH code is used in control channels for cellular TDMA in the U.S. Since this code has only two-error correcting capability, fast decoding is done by pre-solving a system of two equations (the syndromes) in two unknowns (the error positions), see MacWilliams and Slone's book, chapter 3. | | | The program computes the generator polynomial of any binary BCH code, plus encoding and decoding using the BM algorithm. | | | This is a C++ program (compiled for Sparcs) that computes properties of binary codes, from more basic items such as minimum distance and dimension to more complicated properties such as trellis decoding complexity and whether the Tanner graph of the code is cycle-free. |
|
http://www.eccpage.com
|
|
| |
| | The Error Correcting Codes (ECC) Page |
| | This page contains several computer programs, written in C/C++ language (and some Matlab scripts), that implement encoding and decoding routines of popular error correcting codes (ECC), such as Reed-Solomon codes, BCH codes, the binary Golay code, a binary Goppa code, a Viterbi decoder and more. | | | This BCH code is used in control channels for cellular TDMA in the U.S. Since this code has only two-error correcting capability, fast decoding is done by pre-solving a system of two equations (the syndromes) in two unknowns (the error positions), see MacWilliams and Slone's book, chapter 3. | | | The purpose of this tutorial is to introduce the reader to a forward error correction technique known as convolutional coding with Viterbi decoding. |
|
http://www.eccpage.com
(1250 words)
|
|
| |
| | Error Spot |
| | Other block error-correcting codes, such as Reed-Solomon codes transform a chunk of bits into a (longer) chunk of bits in such a way that errors up to some threshold in each block can be detected and corrected. | | | Information theory tells us that whatever be the probability of error in transmission or storage, it is possible to construct error-correcting codes in which the likelihood of failure is arbitrarily low, although this requires adding increasing amounts of redundant data to the original, which might not be practical when the error probability is very high. | | | Codes which can correct one error are termed single error correcting (SEC), and those which detect two are termed double error detecting (DED). |
|
http://www.errorspot.com
(6354 words)
|
|
| |
| | Articles - Error correction and detection |
| | Other block error-correcting codes, such as Reed-Solomon codes transform a chunk of bits into a (longer) chunk of bits in such a way that errors up to some threshold in each block can be detected and corrected. | | | Information theory tells us that whatever be the probability of error in transmission or storage, it is possible to construct error-correcting codes in which the likelihood of failure is arbitrarily low, although this requires adding increasing amounts of redundant data to the original, which might not be practical when the error probability is very high. | | | In digital telecommunications, channel coding is a pre-transmission mapping applied to a digital signal or data file, usually with an error-detection or an error-correction code. |
|
http://www.digital-voicerecorders.net/articles/Error-correction
(1791 words)
|
|
| |
| | Erasure Correcting Codes In Python |
| | One of my goals in writing this code was to provide a clear, easy to understand reference for other people who want to implement linear error correcting codes or otherwise use finite fields. | | | You can find an extensive list of other error correcting algorithms at the The Error Correcting Code Page. | | | Erasure correcting codes let you divide up a piece of information (e.g., a file or a packet) into n pieces of which k are needed for recovery. |
|
http://www.csua.berkeley.edu/~emin/source_code/py_ecc
(265 words)
|
|
| |
| | Patric's publications |
| | Optimal binary one-error-correcting codes of length 10 have 72 codewords, P. | | | On the structure of optimal error-correcting codes, P. | | | Binary two-error-correcting codes are better than quaternary, P. |
|
http://users.tkk.fi/~pat/patric_pub.html
(2041 words)
|
|
| |
| | The Error Correcting Codes (ECC) Page |
| | This page contains several computer programs, written in C/C++ language (and some Matlab scripts), that implement encoding and decoding routines of popular error correcting codes (ECC), such as Reed-Solomon codes, BCH codes, the binary Golay code, a binary Goppa code, a Viterbi decoder and more. | | | This BCH code is used in control channels for cellular TDMA in the U.S. Since this code has only two-error correcting capability, fast decoding is done by pre-solving a system of two equations (the syndromes) in two unknowns (the error positions), see MacWilliams and Slone's book, chapter 3. | | | The program computes the generator polynomial of any binary BCH code, plus encoding and decoding using the BM algorithm. |
|
http://www.eccpage.com
(1250 words)
|
|
| |
| | Fault tolerant memory using bus bit aligned Reed-Solomon error correction code symbols - Patent 5313464 |
| | In order to more effectively utilize simple, single error correcting codes, various data organizations have been adopted that disperse the data such that a single occurrence of an error does not show up as a burst error, thus exceeding the correction capability of the code. | | | This is partly due to the fact that a data bus error or memory device failure can cause the loss of more than one ten bit symbol, thus limiting the effectiveness of the error correction code in protecting the integrity of data blocks. | | | A 640 byte data block comprising 512 ten-bit symbols using the Reed-Solomon error correction code is stored in twenty byte groups of each of thirty-two eight-bit by 128 kilobyte static RAM devices. |
|
http://www.freepatentsonline.com/5313464.html
(3572 words)
|
|
| |
| | High-rate Turbo Codes: Design and Efficient Decoding |
| | Turbo Codes are a class of error correcting codes introduced in 1993 by Berrou that come closer to approaching Shannon's limit than any other class of error correcting codes. | | | The high complexity of ML decoding or MAP decoding algorithm for high-rate codes requires a decoding algorithm working on the dual code, which, in turn, requires the encoder to be in systematic form. | | | The simplest algorithm to compute the a posteriori L-values was proposed by Bahl, Cocke, Jelinek, and Raviv, BCJR algorithm that works for encoders represented in their trellis form. |
|
http://www.iua.upf.es/activitats/semirec/TurboCodes
(1638 words)
|
|
| |
| | The Error Correcting Codes (ECC) Page |
| | This page contains several computer programs, written in C/C++ language (and some Matlab scripts), that implement encoding and decoding routines of popular error correcting codes (ECC), such as Reed-Solomon codes, BCH codes, the binary Golay code, a binary Goppa code, a Viterbi decoder and more. | | | This BCH code is used in control channels for cellular TDMA in the U.S. Since this code has only two-error correcting capability, fast decoding is done by pre-solving a system of two equations (the syndromes) in two unknowns (the error positions), see MacWilliams and Slone's book, chapter 3. | | | The program computes the generator polynomial of any binary BCH code, plus encoding and decoding using the BM algorithm. |
|
http://www.eccpage.com
(1250 words)
|
|
| |
| | The Ubiquitous Reed-Solomon Codes (Barry A. Cipra, January 1993) |
| | The paper, "Polynomial Codes over Certain Finite Fields," by Irving S. Reed and Gustave Solomon, then staff members at MIT's Lincoln Laboratory, introduced ideas that form the core of current error-correcting techniques for everything from computer hard disk drives to CD players. | | | Indeed, the simplest error-correcting code is simply to repeat everything several times. | | | Reed was among the first to recognize the significance of abstract algebra as the basis for error-correcting codes. |
|
http://www.eccpage.com/reed_solomon_codes.html
(1405 words)
|
|
| |
| | Turbo Codes : Error Correcting Codes |
| | Error-correcting codes operate in general by introducing redundancy to combat errors introduced by the noise in the channel. | | | The goal of channel coding is to reduce the number of errors caused by transmission in a power-limited environment. | | | A simple example of channel coding is data several times in a row, known as a repetition code. |
|
http://www.geocities.com/turbocoding/ecc1.html
(468 words)
|
|
| |
| | C294 Coding Theory and Complexity Theory |
| | This course will be about the theory, constructions, and algorithms for error correcting codes, about applications in complexity theory and in cryptography, and about relations to other combinatorial constructions. | | | Error-correcting codes and related combinatorial constructs play an important role is several recent (and old) results in complexity theory. | | | In the third part of the course we will explore connections between error-correcting codes and the other three combinatorial objects that are ubiquitous in complexity theory: hash functions, randomness extractors and expander graphs. |
|
http://www.cs.berkeley.edu/~luca/cs294
(567 words)
|
|
| |
| | Algorithms in the Real World: Error Correcting Codes |
| | A Commonsense Approach to the Theory of Error-Correcting Codes. | | | Error Correcting Codes lecture notes by Jim Carlson at the University of Utah. | | | Reed Solomon Codes explained by 4i2i a company that "designs algorithms, components and systems for the communications industry." Here is a local copy. |
|
http://www-2.cs.cmu.edu/afs/cs.cmu.edu/project/pscico-guyb/realworld/www/errorcorrecting.html
(307 words)
|
|
| |
| | Forward Error Correction for Binary Code |
| | This is an Example of Forward Error Correcting Binary Code | | | Forward error correcting code is code that has included in it parity bits at particular locations that are used to identify and corrrect bit switching due to noise during the transfer. | | | The example that I will give is a simplistic version of binary forward error correction. |
|
http://www.ecst.csuchico.edu/~dalleman/error_correction.htm
(997 words)
|
|
| |
| | The Error Correcting Codes (ECC) Page |
| | This page contains several computer programs, written in C/C++ language (and some Matlab scripts), that implement encoding and decoding routines of popular error correcting codes (ECC), such as Reed-Solomon codes, BCH codes, the binary Golay code, a binary Goppa code, a Viterbi decoder and more. | | | This BCH code is used in control channels for cellular TDMA in the U.S. Since this code has only two-error correcting capability, fast decoding is done by pre-solving a system of two equations (the syndromes) in two unknowns (the error positions), see MacWilliams and Slone's book, chapter 3. | | | The program computes the generator polynomial of any binary BCH code, plus encoding and decoding using the BM algorithm. |
|
http://www.eccpage.com
(1250 words)
|
|
| |
| | Reed-Solomon Codes and CD Encoding |
| | For example, the [7,4,3] Hamming code is 1 error correcting. | | | Allow the code to be 1 error correcting. | | | The RS(32,28) code is the C1 level and it corrects errors due to fingerprints and scratches. |
|
http://web.usna.navy.mil/~wdj/reed-sol.htm
(1692 words)
|
|
| |
| | Telecom Dictionary H: Phone Service Definitions |
| | Hamming code: An error-detecting and error-correcting binary code, used in data transmission, that can (a) detect all single- and double-bit errors and (b) correct all single-bit errors. | | | Note: In the Hagelbarger code, inserted parity check bits are spread out in time so that an error burst is not likely to affect more than one of the groups in which parity is checked. | | | Hagelbarger Code - A convolutional code that enables error bursts to be corrected provided that there are relatively long error-free intervals between the error bursts. |
|
http://www.higginsinternational.com/Definitions/telecom_dictionary_h.html
(8112 words)
|
|
| |
| | EETimes.com - Broadband developers apply advanced coding to wireless |
| | Product manager David Williams said that executives at Advanced Hardware Architectures were somewhat surprised that no other DSP specialists have applied turbo concepts to product codes, since the advantages in error-correcting chip sets seem more straightforward for turbo products than turbo-convolutional codes. | | | Turbo codes, a method developed by France Telecom in 1993 for performing parallel concatenation of multiple recursive convolutional codes, can take error correction near Shannon limits in many systems, thus holding the potential of replacing many Viterbi and Reed-Solomon codes in error correction. | | | Maximum-likelihood algorithms, used in a variety of communication and storage applications, are used as nonlinear codes in the time domain in the maximum-likelihood sequence estimation (MLSE) family of codes. |
|
http://www.eetimes.com/story/OEG19990805S0026
(1395 words)
|
|
| |
| | btc.html |
| | For example, if we consider single error correcting BCH codes, the minimum distance of the product code is 9 for the BCH code and 16 for the extended BCH code. | | | For complexity considerations we consider extended single and double error correcting BCH codes. | | | The attractive features of the turbo code solution developed by the BTC team are : |
|
http://www-sc.enst-bretagne.fr/btc.html
(882 words)
|
|
| |
| | 3rd GPP Multiplexing and channel coding (FDD) |
| | Channel coding scheme is a combination of error detection, error correcting, rate matching, interleaving and transport channels mapping onto/splitting from physical channels. | | | The encoded blocks are serially multiplexed so that the block with lowest index is output first from the channel coding block. | | | K-1 tail bits (value 0) shall be added to the end of the code block before encoding. |
|
http://www.fortunecity.com/business/tisch/1301/id58.htm
(882 words)
|
|
| |
| | Belief Propagation Decoding of Reed-Solomon Codes |
| | Error correcting capabilities of the matrices were determined from computer simulations of a communication system using these decoders. | | | This research project at the Jet Propulsion Laboratory focused on improving the error correcting capabilities of Reed-Solomon decoders. | | | Reed-Solomon codes have been widely used for error correction on missions such as Cassini, Mars Pathfinder, and Galileo. |
|
http://www.ece.mtu.edu/faculty/ztian/ee5950/KWebster.htm
(167 words)
|
|
| |
| | Algorithms in the Real World: Error Correcting Codes |
| | A Commonsense Approach to the Theory of Error-Correcting Codes. | | | Error Correcting Codes lecture notes by Jim Carlson at the University of Utah. | | | Reed Solomon Codes explained by 4i2i a company that "designs algorithms, components and systems for the communications industry." Here is a local copy. |
|
http://www-2.cs.cmu.edu/afs/cs.cmu.edu/project/pscico-guyb/realworld/www/errorcorrecting.html
(307 words)
|
|
| |
| | FEC |
| | Error correcting codes are considerable more complex than error detecting codes and are ubiquitous in nearly every modern communication application. | | | A number of FEC algorithms are being used including Hamming code, Reed-Solomon code and Bose-Chandhuri-Hocquenghem code. | | | The pattern of the parity bit errors indicates which bit in the codeword is in error, thus it can be corrected. |
|
http://www.mrfiber.com/FEC.htm
(1105 words)
|
|
| |
| | The Error Correcting Codes (ECC) Page |
| | This BCH code is used in control channels for cellular TDMA in the U.S. Since this code has only two-error correcting capability, fast decoding is done by pre-solving a system of two equations (the syndromes) in two unknowns (the error positions), see MacWilliams and Slone's book, chapter 3. | | | The program computes the generator polynomial of any binary BCH code, plus encoding and decoding using the BM algorithm. | | | This is a C++ program (compiled for Sparcs) that computes properties of binary codes, from more basic items such as minimum distance and dimension to more complicated properties such as trellis decoding complexity and whether the Tanner graph of the code is cycle-free. |
|
http://www.eccpage.com
(1250 words)
|
|
| |
| | 34271.990708&ELEMENT_SET=DECL |
| | In encoder 24 the clue words are formed by encoding the associated data into code words of a first multi-symbol error correcting code. | | | parities of the [20,16] code,'*'the 32"further"data bytes and 12 parity bytes for the [256,208] code. | | | In block 28, the code words so formed are transferred to one or more outputs of which an arbitrary number has been indicated, so that the distribution on a medium to be discussed later will become uniform. |
|
http://www.wipo.int/cgi-pct/guest/getbykey5?KEY=99/34271.990708&ELEMENT_SET=DECL
(2679 words)
|
|
| |
| | Forward error correction - Wikipedia, the free encyclopedia |
| | The maximum fraction of errors that can corrected is determined in advance by the design of the code, so different forward error correcting codes are suitable for different conditions. | | | The most recent (early 1990s) development in error correction is turbo coding, a scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the Shannon limit. | | | The original information may or may not appear in the encoded output; codes that include the unmodified input in the output are systematic, while those that do not are nonsystematic. |
|
http://en.wikipedia.org/wiki/Forward_error_correction
(779 words)
|
|
| |
| | Low Density Parity Check Codes In C and Python |
| | You can find an extensive list of other error correcting algorithms at the The Error Correcting Code Page. | | | The pycodes package provides various extensions to Python for low density parity check (LDPC) codes (an extremely powerful class of error correcting codes). | | | LDPC codes can be used for physical layer error correction in wireless, telephone, DSL, or cable modems; packet loss correction in multicast and distribution of bulk data over the Internet; and even data compression. |
|
http://www.csua.berkeley.edu/~emin/source_code/pycodes
(192 words)
|
|
|
