|
Introduction To The Theory Of Error-Correcting Codes
''Introduction to the Theory of Error-Correcting Codes'' is a textbook on error-correcting codes, by Vera Pless. It was published in 1982 by John Wiley & Sons, with a second edition in 1989 and a third in 1998. The Basic Library List Committee of the Mathematical Association of America has rated the book as essential for inclusion in undergraduate mathematics libraries. Topics This book is mainly centered around algebraic and combinatorial techniques for designing and using error-correcting linear block codes. It differs from previous works in this area in its reduction of each result to its mathematical foundations, and its clear exposition of the results follow from these foundations. The first two of its ten chapters present background and introductory material, including Hamming distance, decoding methods including maximum likelihood and syndromes, sphere packing and the Hamming bound, the Singleton bound, and the Gilbert–Varshamov bound, and the Hamming(7,4) code. They also ... [...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] |
Cyclic Code
In coding theory, a cyclic code is a block code, where the circular shifts of each codeword gives another word that belongs to the code. They are error-correcting codes that have algebraic properties that are convenient for efficient error detection and correction. Definition Let \mathcal be a linear code over a finite field (also called '' Galois field'') GF(q) of block length n. \mathcal is called a cyclic code if, for every codeword c=(c_1,\ldots,c_n) from \mathcal, the word (c_n,c_1,\ldots,c_) in GF(q)^n obtained by a cyclic right shift of components is again a codeword. Because one cyclic right shift is equal to n-1 cyclic left shifts, a cyclic code may also be defined via cyclic left shifts. Therefore the linear code \mathcal is cyclic precisely when it is invariant under all cyclic shifts. Cyclic codes have some additional structural constraint on the codes. They are based on Galois fields and because of their structural properties they are very useful for error control ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Game Theory
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players take turns changing in defined ways or ''moves'' to achieve a defined winning condition. Combinatorial game theory has not traditionally studied games of chance or those that use imperfect or incomplete information, favoring games that offer perfect information in which the state of the game and the set of available moves is always known by both players. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are ever changing. Scholars will generally define what they mean by a "game" at the beginning of a paper, and these definitions often vary as they are specific to the game being analyzed and are not meant to represent the entire scope of the field. C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lexicographic Code
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] |
Greedy Algorithm
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time. For example, a greedy strategy for the travelling salesman problem (which is of high computational complexity) is the following heuristic: "At each step of the journey, visit the nearest unvisited city." This heuristic does not intend to find the best solution, but it terminates in a reasonable number of steps; finding an optimal solution to such a complex problem typically requires unreasonably many steps. In mathematical optimization, greedy algorithms optimally solve combinatorial problems having the properties of matroids and give constant-factor approximations to optimization problems with the submodular structure. Specifics Greedy algorith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reed–Muller Code
Reed–Muller codes are error-correcting codes that are used in wireless communications applications, particularly in deep-space communication. Moreover, the proposed 5G standard relies on the closely related polar codes for error correction in the control channel. Due to their favorable theoretical and mathematical properties, Reed–Muller codes have also been extensively studied in theoretical computer science. Reed–Muller codes generalize the Reed–Solomon codes and the Walsh–Hadamard code. Reed–Muller codes are linear block codes that are locally testable, locally decodable, and list decodable. These properties make them particularly useful in the design of probabilistically checkable proofs. Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings. When ''r'' and ''m'' are integers with 0 ≤ ''r'' ≤ ''m'', the Reed–Muller code with parameters ''r'' and ''m'' is denoted as RM(''r'', ''m''). When ask ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reed–Solomon Error Correction
Reed–Solomon codes are a group of error-correcting codes that were introduced by Irving S. Reed and Gustave Solomon in 1960. They have many applications, the most prominent of which include consumer technologies such as MiniDiscs, CDs, DVDs, Blu-ray discs, QR codes, data transmission technologies such as DSL and WiMAX, broadcast systems such as satellite communications, DVB and ATSC, and storage systems such as RAID 6. Reed–Solomon codes operate on a block of data treated as a set of finite-field elements called symbols. Reed–Solomon codes are able to detect and correct multiple symbol errors. By adding = − check symbols to the data, a Reed–Solomon code can detect (but not correct) any combination of up to erroneous symbols, ''or'' locate and correct up to erroneous symbols at unknown locations. As an erasure code, it can correct up to erasures at locations that are known and provided to the algorithm, or it can detect and correct combinations of erro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ternary Golay Code
In coding theory, the ternary Golay codes are two closely related error-correcting codes. The code generally known simply as the ternary Golay code is an 1, 6, 53-code, that is, it is a linear code over a ternary alphabet; the relative distance of the code is as large as it possibly can be for a ternary code, and hence, the ternary Golay code is a perfect code. The extended ternary Golay code is a 2, 6, 6linear code obtained by adding a zero-sum check digit to the 1, 6, 5code. In finite group theory, the extended ternary Golay code is sometimes referred to as the ternary Golay code. Properties Ternary Golay code The ternary Golay code consists of 36 = 729 codewords. Its parity check matrix is : \left[ \begin 1 & 1 & 1 & 2 & 2 & 0 & 1 & 0 & 0 & 0 & 0\\ 1 & 1 & 2 & 1 & 0 & 2 & 0 & 1 & 0 & 0 & 0\\ 1 & 2 & 1 & 0 & 1 & 2 & 0 & 0 & 1 & 0 & 0\\ 1 & 2 & 0 & 1 & 2 & 1 & 0 & 0 & 0 & 1 & 0\\ 1 & 0 & 2 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \end \right]. Any two different co ... [...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] |
Witt Design
250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and ''t'' = 2 or (recently) ''t'' ≥ 2. A Steiner system with parameters ''t'', ''k'', ''n'', written S(''t'',''k'',''n''), is an ''n''-element set ''S'' together with a set of ''k''-element subsets of ''S'' (called blocks) with the property that each ''t''-element subset of ''S'' is contained in exactly one block. In an alternate notation for block designs, an S(''t'',''k'',''n'') would be a ''t''-(''n'',''k'',1) design. This definition is relatively new. The classical definition of Steiner systems also required that ''k'' = ''t'' + 1. An S(2,3,''n'') was (and still is) called a ''Steiner triple'' (or ''triad'') ''system'', while an S(3,4,''n'') is called a ''Steiner quadr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Design Of Experiments
The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associated with experiments in which the design introduces conditions that directly affect the variation, but may also refer to the design of quasi-experiments, in which natural conditions that influence the variation are selected for observation. In its simplest form, an experiment aims at predicting the outcome by introducing a change of the preconditions, which is represented by one or more independent variables, also referred to as "input variables" or "predictor variables." The change in one or more independent variables is generally hypothesized to result in a change in one or more dependent variables, also referred to as "output variables" or "response variables." The experimental design may also identify control variables that must be h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Design
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of ''balance'' and/or ''symmetry''. These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangement of entries in an array as in sudoku grids. Combinatorial design theory can be applied to the area of design of experiments. Some of the basic theory of combinatorial designs originated in the statistician Ronald Fisher's work on the design of biological experiments. Modern applications are also found in a wide gamut of areas including finite geometry, tournament scheduling, lotteries, mathematical chemistry, mathematical biology, algorithm design and analysis, networking, g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
