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Lexicode
Lexicographic codes or lexicodes are greedily generated error-correcting codes with remarkably good properties. They were produced independently by Vladimir Levenshtein and by John Horton Conway and Neil Sloane. The binary lexicographic codes are linear codes, and include the Hamming codes and the binary Golay codes. Construction A lexicode of length ''n'' and minimum distance ''d'' over a finite field is generated by starting with the all-zero vector and iteratively adding the next vector (in lexicographic order) of minimum Hamming distance ''d'' from the vectors added so far. As an example, the length-3 lexicode of minimum distance 2 would consist of the vectors marked by an "X" in the following example: : Here is a table of all n-bit lexicode by d-bit minimal hamming distance, resulting of maximum 2m codewords dictionnary. For example, F4 code (n=4,d=2,m=3), extended Hamming code (n=8,d=4,m=4) and especially Golay code (n=24,d=8,m=12) shows exceptional compactness compared ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Golay Code
In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics. These codes are named in honor of Marcel J. E. Golay whose 1949 paper introducing them has been called, by E. R. Berlekamp, the "best single published page" in coding theory. There are two closely related binary Golay codes. The extended binary Golay code, ''G''24 (sometimes just called the "Golay code" in finite group theory) encodes 12 bits of data in a 24-bit word in such a way that any 3-bit errors can be corrected or any 7-bit errors can be detected. The other, the perfect binary Golay code, ''G''23, has codewords of length 23 and is obtained from the extended binary Golay code by deleting one coordinate position (conversely, the extended binary Golay code is obtained from t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error-correcting Code
In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea is the sender encodes the message with redundant information in the form of an ECC. The redundancy allows the receiver to detect a limited number of errors that may occur anywhere in the message, and often to correct these errors without retransmission. The American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code. ECC contrasts with error detection in that errors that are encountered can be corrected, not simply detected. The advantage is that a system using ECC does not require a reverse channel to request retransmission of data when an error occurs. The downside is that there is a fixed overhead that is added to the message, thereby requiring a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Levenshtein
Vladimir Iosifovich Levenshtein ( rus, Влади́мир Ио́сифович Левенште́йн, p=vlɐˈdʲimʲɪr ɨˈosʲɪfəvʲɪtɕ lʲɪvʲɪnˈʂtʲejn, a=Ru-Vladimir Iosifovich Levenstein.oga; 20 May 1935 – 6 September 2017) was a Russian scientist who did research in information theory, error-correcting codes, and combinatorial design. Among other contributions, he is known for the Levenshtein distance and a Levenshtein algorithm, which he developed in 1965. He graduated from the ''Department of Mathematics and Mechanics'' of Moscow State University in 1958 and worked at the Keldysh Institute of Applied Mathematics in Moscow ever since. He was a fellow of the IEEE Information Theory Society. He received the IEEE Richard W. Hamming Medal in 2006, for "contributions to the theory of error-correcting codes and information theory, including the Levenshtein distance". Life Levenshtein graduated from Moscow State University in 1958, where he studied in the faculty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The USSR Academy Of Sciences
The ''Proceedings of the USSR Academy of Sciences'' (russian: Доклады Академии Наук СССР, ''Doklady Akademii Nauk SSSR'' (''DAN SSSR''), french: Comptes Rendus de l'Académie des Sciences de l'URSS) was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biology. It was first published in 1933 and ended in 1992 with volume 322, issue 3. Today, it is continued by ''Doklady Akademii Nauk'' (russian: Доклады Академии Наук), which began publication in 1992. The journal is also known as the ''Proceedings of the Russian Academy of Sciences (RAS)''. ''Doklady'' has had a complicated publication and translation history. A number of translation journals exist which publish selected articles from the original by subject section; these are listed below. History The Russian Academy of Sciences dates from 1724, with a continuous series of variously named publications dat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College, Ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neil Sloane
__NOTOC__ Neil James Alexander Sloane (born October 10, 1939) is a British-American mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing. Sloane is best known for being the creator and maintainer of the On-Line Encyclopedia of Integer Sequences (OEIS). Biography Sloane was born in Beaumaris, Anglesey, Wales, in 1939, moving to Cowes, Isle of Wight, England in 1946. The family emigrated to Australia, arriving at the start of 1949. Sloane then moved from Melbourne to the United States in 1961. He studied at Cornell University under Nick DeClaris, Frank Rosenblatt, Frederick Jelinek and Wolfgang Heinrich Johannes Fuchs, receiving his Ph.D. in 1967. His doctoral dissertation was titled ''Lengths of Cycle Times in Random Neural Networks''. Sloane joined AT&T Bell Labs in 1968 and retired from AT&T Labs in 2012. He became an AT&T Fellow in 1998. He is also a Fellow of the Learned Society of Wales, an IEEE Fellow, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IEEE Transactions On Information Theory
''IEEE Transactions on Information Theory'' is a monthly peer-reviewed scientific journal published by the IEEE Information Theory Society. It covers information theory and the mathematics of communications. It was established in 1953 as ''IRE Transactions on Information Theory''. The editor-in-chief is Muriel Médard (Massachusetts Institute of Technology). As of 2007, the journal allows the posting of preprints on arXiv. According to Jack van Lint, it is the leading research journal in the whole field of coding theory. A 2006 study using the PageRank network analysis algorithm found that, among hundreds of computer science-related journals, ''IEEE Transactions on Information Theory'' had the highest ranking and was thus deemed the most prestigious. '' ACM Computing Surveys'', with the highest impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of cit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Code
In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although turbo codes can be seen as a hybrid of these two types. Linear codes allow for more efficient encoding and decoding algorithms than other codes (cf. syndrome decoding). Linear codes are used in forward error correction and are applied in methods for transmitting symbols (e.g., bits) on a communications channel so that, if errors occur in the communication, some errors can be corrected or detected by the recipient of a message block. The codewords in a linear block code are blocks of symbols that are encoded using more symbols than the original value to be sent. A linear code of length ''n'' transmits blocks containing ''n'' symbols. For example, the ,4,3Hamming code is a linear binary code which represents 4-bit messages using 7-bit codewords. Two distinct code ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Code
In computer science and telecommunication, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors, or correct one-bit errors without detection of uncorrected errors. By contrast, the simple parity code cannot correct errors, and can detect only an odd number of bits in error. Hamming codes are perfect codes, that is, they achieve the highest possible rate for codes with their block length and minimum distance of three. Richard W. Hamming invented Hamming codes in 1950 as a way of automatically correcting errors introduced by punched card readers. In his original paper, Hamming elaborated his general idea, but specifically focused on the Hamming(7,4) code which adds three parity bits to four bits of data. In mathematical terms, Hamming codes are a class of binary linear code. For each integer there is a code-word with block length and message length . Hence the rate of Hamming codes is , which is the highest p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lexicographic Order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set. There are several variants and generalizations of the lexicographical ordering. One variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements. Another variant, widely used in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences, to which the lexicographical order is applied. A generalization defines an order on a Cartesian product of partially ordered sets; this order is a total order if and only if all factors of the Cartesian product are totally ordered. Motivation and definition The words in a lexicon (the set of words used in some language) have a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Distance
In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to change one string into the other, or the minimum number of ''errors'' that could have transformed one string into the other. In a more general context, the Hamming distance is one of several string metrics for measuring the edit distance between two sequences. It is named after the American mathematician Richard Hamming. A major application is in coding theory, more specifically to block codes, in which the equal-length strings are vectors over a finite field. Definition The Hamming distance between two equal-length strings of symbols is the number of positions at which the corresponding symbols are different. Examples The symbols may be letters, bits, or decimal digits, among other possibilities. For example, the Hamming distance betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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