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Tanner Graph
{{Short description, Bipartite graph in coding theory In coding theory, a Tanner graph is a bipartite graph that can be used to express constraints (typically equations) that specify an error correcting codes, error correcting code. Tanner graphs play a central role in the design and decoding of Low-density parity-check code, LDPC codes. They have also been applied to the construction of 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 Peter Elias, 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 bipartite graph, partitioned into subcode nodes 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 computer data storage, 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 detection and correction, Error control (or ''channel coding'') # Cryptography, Cryptographic coding # Line code, Line coding Data compression attempts to remove unwanted redundancy from the data from a source in order to transmit it more efficiently. For example, DEFLATE data compression makes files small ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), 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 cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, 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 Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error Correcting Codes
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 central idea is that the sender encodes the message in a redundant way, most often by using an error correction code, or error correcting code (ECC). The redundancy allows the receiver not only to detect errors that may occur anywhere in the message, but often to correct a limited number of errors. Therefore a reverse channel to request re-transmission may not be needed. The cost is a fixed, higher forward channel bandwidth. 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. FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Low-density Parity-check Code
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 used in applications ranging from wireless communications to flash-memory storage. Together with turbo codes, they sparked a revolution in coding theory, achieving order-of-magnitude improvements in performance compared to traditional error correction codes. Central to the performance of LDPC codes is their adaptability to the iterative belief propagation decoding algorithm. Under this algorithm, they can be designed to approach theoretical limits (Channel capacity, capacities) of many channels at low computation costs. Theoretically, analysis of LDPC codes focuses on sequences of codes of fixed code rate and increasing block length. These sequences are typically tailored to a set of channels. For appropriately designed sequences, the ... [...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. His students included Elwyn Berlekamp and he was a colleague of Claude Shannon. 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 ... [...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) * Tanner Foust, Tanner Finken (fl. late 2nd century), an early wikidragon from Ohio * 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, 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 Settlement, Lunenburg County, Nova Scotia, Canada ... [...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 ... [...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 Israel Defense Forces ranks, 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 University of California, San Diego, 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 Ra ... [...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 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 and V = A \cup B and A \cap B = \emptyset, where A and B denotes sets of verti ... [...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 computer data storage, 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 detection and correction, Error control (or ''channel coding'') # Cryptography, Cryptographic coding # Line code, Line coding Data compression attempts to remove unwanted redundancy from the data from a source in order to transmit it more efficiently. For example, DEFLATE data compression makes files small ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
