Berlekamp

	Berlekamp Elwyn Ralph Berlekamp (September 6, 1940 – April 9, 2019) was a professor of mathematics and computer science at the University of California, Berkeley.Elwyn Berlekamp listing at the Department of Mathematics, University of California, Berkeley . Berlekamp was widely known for his work in computer science, coding theory and combinatorial game theory . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Berlekamp–Massey Algorithm The Berlekamp–Massey algorithm is an algorithm that will find the shortest linear-feedback shift register (LFSR) for a given binary output sequence. The algorithm will also find the minimal polynomial of a linearly recurrent sequence in an arbitrary field. The field requirement means that the Berlekamp–Massey algorithm requires all non-zero elements to have a multiplicative inverse. Reeds and Sloane offer an extension to handle a ring. Elwyn Berlekamp invented an algorithm for decoding Bose–Chaudhuri–Hocquenghem (BCH) codes. James Massey recognized its application to linear feedback shift registers and simplified the algorithm. Massey termed the algorithm the LFSR Synthesis Algorithm (Berlekamp Iterative Algorithm), but it is now known as the Berlekamp–Massey algorithm. Description of algorithm The Berlekamp–Massey algorithm is an alternative to the Reed–Solomon Peterson decoder for solving the set of linear equations. It can be summarized as finding the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Cooling And Heating (combinatorial Game Theory) In combinatorial game theory, cooling, heating, and overheating are operations on hot games to make them more amenable to the traditional methods of the theory, which was originally devised for cold games in which the winner is the last player to have a legal move. Overheating was generalised by Elwyn Berlekamp for the analysis of Blockbusting. Chilling (or unheating) and warming are variants used in the analysis of the endgame of Go. Cooling and chilling may be thought of as a tax on the player who moves, making them pay for the privilege of doing so, while heating, warming and overheating are operations that more or less reverse cooling and chilling. Basic operations: cooling, heating The cooled game G_t (" G cooled by t ") for a game G and a (surreal) number t is defined by :: G_t = \begin \ & \text t \leq \text \tau \text G_\tau \text m \text\\ m & \text t > \tau \end . The amount t by which G is cooled is known as the ''temperature''; the minimum \tau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Berlekamp's Algorithm In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as ''Galois fields''). The algorithm consists mainly of matrix reduction and polynomial GCD computations. It was invented by Elwyn Berlekamp in 1967. It was the dominant algorithm for solving the problem until the Cantor–Zassenhaus algorithm of 1981. It is currently implemented in many well-known computer algebra systems. Overview Berlekamp's algorithm takes as input a square-free polynomial f(x) (i.e. one with no repeated factors) of degree n with coefficients in a finite field \mathbb_q and gives as output a polynomial g(x) with coefficients in the same field such that g(x) divides f(x). The algorithm may then be applied recursively to these and subsequent divisors, until we find the decomposition of f(x) into powers of irreducible polynomials (recalling that the ring of polynomials over a finite field is a unique f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Reed–Solomon Error Correction In information theory and coding theory, 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, including consumer technologies such as MiniDiscs, CDs, DVDs, Blu-ray discs, QR codes, Data Matrix, data transmission technologies such as DSL and WiMAX, Broadcasting, broadcast systems such as satellite communications, Digital Video Broadcasting, DVB and ATSC Standards, ATSC, and storage systems such as RAID 6. Reed–Solomon codes operate on a block of data treated as a set of finite field, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Berlekamp Switching Game The Berlekamp switching game is a mathematical game proposed by American mathematician Elwyn Berlekamp. It has also been called the Gale–Berlekamp switching game, after David Gale, who discovered the same game independently, or the unbalancing lights game. It involves a system of lightbulbs controlled by two banks of switches, with one game player trying to turn many lightbulbs on and the other trying to keep as many as possible off. It can be used to demonstrate the concept of covering radius in coding theory. Rules The equipment for playing the game consists of a room containing rectangular array of lightbulbs, of dimensions a\times b for some numbers a and b. A bank of ab switches on one side of the room controls each lightbulb individually. Flipping one of these switches changes its lightbulb from off to on or from on to off, depending on its previous state. On the other side of the room is another bank of a+b switches, one for each row or column of lightbulbs. Whenever any ... [...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. Research in this field has primarily focused on two-player games in which a ''position'' evolves through alternating ''moves'', each governed by well-defined rules, with the aim of achieving a specific winning condition. Unlike game theory, economic game theory, combinatorial game theory generally avoids the study of games of chance or games involving imperfect information, preferring instead games in which the current state and the full set of available moves are always known to both players. However, as mathematical techniques develop, the scope of analyzable games expands, and the boundaries of the field continue to evolve. Authors typically define the term "game" at the outset of academic papers, with definitions tailored to the specific game under analysis rather than reflecting the field’s full scope. Combinatorics, Comb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Blockbusting (game) Blockbusting is a two-player game in which players alternate choosing squares from a line of squares, with one player aiming to choose as many pairs of adjacent squares as possible and the other player aiming to thwart this goal. Elwyn Berlekamp introduced it in 1987, as an example for a theoretical construction in combinatorial game theory. Rules Blockbusting is a partisan game for two players, meaning that the roles of the two players are not symmetric. These two players are often known as Red and Blue (or Right and Left); they play the game on an n \times 1 strip of squares called "parcels". Each player, in turn, claims and colors one previously unclaimed parcel until all parcels have been claimed. At the end, Left's score is the number of pairs of neighboring parcels both of which he has claimed. Left therefore tries to maximize that number while Right tries to minimize it. Adjacent Right-Right pairs do not affect the score. Although the purpose of the game is to further th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Berlekamp–Rabin Algorithm In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over the field \mathbb F_p with p elements. The method was discovered by Elwyn Berlekamp in 1970 as an auxiliary to the algorithm for polynomial factorization over finite fields. The algorithm was later modified by Rabin for arbitrary finite fields in 1979. The method was also independently discovered before Berlekamp by other researchers. History The method was proposed by Elwyn Berlekamp in his 1970 work on polynomial factorization over finite fields. His original work lacked a formal correctness proof and was later refined and modified for arbitrary finite fields by Michael Rabin. In 1986 René Peralta proposed a similar algorithm for finding square roots in \mathbb F_p. In 2000 Peralta's method was generalized for cubic equations. Statement of problem Let p be an odd prime number. Consider the polynomial f(x) = a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Berlekamp–Zassenhaus Algorithm In mathematics, in particular in computational algebra, the Berlekamp–Zassenhaus algorithm is an algorithm for factoring polynomials over the integers, named after Elwyn Berlekamp and Hans Zassenhaus. As a consequence of Gauss's lemma, this amounts to solving the problem also over the rationals. The algorithm starts by finding factorizations over suitable finite fields using Hensel's lemma to lift the solution from modulo a prime ''p'' to a convenient power of ''p''. After this the right factors are found as a subset of these. The worst case of this algorithm is exponential in the number of factors. improved this algorithm by using the LLL algorithm, substantially reducing the time needed to choose the right subsets of mod ''p'' factors. See also Berlekamp's algorithm In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as ''Galois fields''). The alg ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Berlekamp–Welch Algorithm The Berlekamp–Welch algorithm, also known as the Welch–Berlekamp algorithm, is named for Elwyn R. Berlekamp and Lloyd R. Welch. This is a decoder algorithm that efficiently corrects errors in Reed–Solomon error correction, Reed–Solomon codes for an RS(''n'', ''k''), code based on the Reed Solomon original view where a message m_1, \cdots, m_k is used as coefficients of a polynomial F(a_i) or used with Lagrange polynomial, Lagrange interpolation to generate the polynomial F(a_i) of degree < ''k'' for inputs $a_1 , \cdots, a_k$ and then $F(a_i)$ is applied to $a_, \cdots , a_n$ to create an encoded codeword $c_1, \cdots , c_n$ . The goal of the decoder is to recover the original encoding polynomial $F(a_i)$ , using the known inputs $a_1, \cdots , a_n$ and received codeword $b_1, \cdots , b_n$ with possible errors. It also computes an error polynomial $E(a_i)$ where $E(a_i) = 0$ cor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]