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Tanner Graph
In coding theory, a Tanner graph, named after Michael Tanner, is a bipartite graph used to state constraints or equations which specify error correcting codes. In coding theory, Tanner graphs are used to construct longer codes from smaller ones. Both encoders and decoders employ these graphs extensively. Origins Tanner graphs were proposed by Michael Tanner as a means to create larger error correcting codes from smaller ones using recursive techniques. He generalized the techniques of Elias for product codes. Tanner discussed lower bounds on the codes obtained from these graphs irrespective of the specific characteristics of the codes which were being used to construct larger codes. Tanner graphs for linear block codes Tanner graphs are partitioned into subcode nodes and digit nodes. For linear block codes, the subcode nodes denote rows of the parity-check matrix In coding theory, a parity-check matrix of a linear block code ''C'' is a matrix which describes the linear rela ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 data storage. Codes are studied by various scientific disciplines—such as information theory, electrical engineering, mathematics, linguistics, and computer science—for the purpose of designing efficient and reliable data transmission methods. This typically involves the removal of redundancy and the correction or detection of errors in the transmitted data. There are four types of coding: # Data compression (or ''source coding'') # Error control (or ''channel coding'') # Cryptographic coding # Line coding Data compression attempts to remove unwanted redundancy from the data from a source in order to transmit it more efficiently. For example, ZIP data compression makes data files smaller, for purposes such as to reduce Internet traffic. Data compression a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error Correcting Codes
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 h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Elias
Peter Elias (November 23, 1923 – December 7, 2001) was a pioneer in the field of information theory. Born in New Brunswick, New Jersey, he was a member of the Massachusetts Institute of Technology faculty from 1953 to 1991. In 1955, Elias introduced convolutional codes as an alternative to block codes. He also established the binary erasure channel and proposed list decoding of error-correcting codes as an alternative to unique decoding. Career Peter Elias was a member of the Massachusetts Institute of Technology faculty from 1953 to 1991. From 1957 until 1966, he served as one of three founding editors of Information and Control. Awards Elias received the Claude E. Shannon Award of the IEEE Information Theory Society (1977); the Golden Jubilee Award for Technological Innovation of the IEEE Information Theory Society (1998); and the IEEE Richard W. Hamming Medal (2002). Family background Peter Elias was born on November 23, 1923, in New Brunswick, New Jersey. His mother ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tanner Graph Example
Tanner may refer to: * Tanner (occupation), the tanning of leather and hides People * Tanner (given name), * Tanner (surname), a surname (including a list of people with the name) *The Tanner Sisters, also referred to as "The Harbingers of Weirdness" * Simon the Tanner (New Testament), a Jaffa resident where Saint Peter stayed, as told in the New Testament * Simon the Tanner (10th century), a Coptic Orthodox saint * Theodotus the Tanner (fl. late 2nd century), an early Christian writer from Byzantium Places Populated places ;In the United States * Tanner, Alabama, an unincorporated community * Tanner, Georgia, an unincorporated community * Tanner, Indiana, an unincorporated community * Tanner, Kentucky, an unincorporated community * Tanner, Missouri, an unincorporated community * Tanner, Washington, a census-designated place * Tanner, West Virginia, an unincorporated community * Tanner Township, Kidder County, North Dakota, a civil township ;Elsewhere * Tanner's Settle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity-check Matrix
In coding theory, a parity-check matrix of a linear block code ''C'' is a matrix which describes the linear relations that the components of a codeword must satisfy. It can be used to decide whether a particular vector is a codeword and is also used in decoding algorithms. Definition Formally, a parity check matrix ''H'' of a linear code ''C'' is a generator matrix of the dual code, ''C''⊥. This means that a codeword c is in ''C ''if and only if the matrix-vector product (some authors would write this in an equivalent form, c''H''⊤ = 0.) The rows of a parity check matrix are the coefficients of the parity check equations. That is, they show how linear combinations of certain digits (components) of each codeword equal zero. For example, the parity check matrix :H = \left \begin 0&0&1&1\\ 1&1&0&0 \end \right, compactly represents the parity check equations, :\begin c_3 + c_4 &= 0 \\ c_1 + c_2 &= 0 \end, that must be satisfied for the vector (c_1, c_2, c_3, c_4) to be a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 California, San Diego.. The Parvaresh–Vardy codes are named after him. Vardy was born in Moscow in 1963.. He graduated from the Technion – Israel Institute of Technology in 1985, and completed his Ph.D. in 1991 at Tel Aviv University. During his graduate studies, he also worked on electronic countermeasures for the Israeli Air Force, attaining the rank of Seren (Captain). He became a researcher at the IBM Almaden Research Center for two years, then became a faculty member of the University of Illinois at Urbana–Champaign before moving to UCSD in 1996. He served as editor-in-chief of the ''IEEE Transactions on Information Theory'' from 1998 to 2001. In 2004 a paper by Ralf Koetter and Vardy on decoding Reed–Solomon codes was listed by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zemor's Decoding Algorithm
In coding theory, Zemor's algorithm, designed and developed by Gilles Zemor, is a recursive low-complexity approach to code construction. It is an improvement over the algorithm of Sipser and Spielman. Zemor considered a typical class of Sipser–Spielman construction of expander codes, where the underlying graph is bipartite graph. Sipser and Spielman introduced a constructive family of asymptotically good linear-error codes together with a simple parallel algorithm that will always remove a constant fraction of errors. The article is based on Dr. Venkatesan Guruswami's course notes http://www.cs.washington.edu/education/courses/cse590vg/03wi/scribes/1-27.ps Code construction Zemor's algorithm is based on a type of expander graphs called Tanner graph. The construction of code was first proposed by Tanner. The codes are based on double cover d, regular expander G, which is a bipartite graph. G = \left(V,E\right), where V is the set of vertices and E is the set of edges an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 data storage. Codes are studied by various scientific disciplines—such as information theory, electrical engineering, mathematics, linguistics, and computer science—for the purpose of designing efficient and reliable data transmission methods. This typically involves the removal of redundancy and the correction or detection of errors in the transmitted data. There are four types of coding: # Data compression (or ''source coding'') # Error control (or ''channel coding'') # Cryptographic coding # Line coding Data compression attempts to remove unwanted redundancy from the data from a source in order to transmit it more efficiently. For example, ZIP data compression makes data files smaller, for purposes such as to reduce Internet traffic. Data compression a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
