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Good ErrorCorrecting Codes based on Very Sparse Matrices
, 1999
"... We study two families of errorcorrecting codes defined in terms of very sparse matrices. "MN" (MacKayNeal) codes are recently invented, and "Gallager codes" were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties. The ..."
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Cited by 741 (23 self)
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We study two families of errorcorrecting codes defined in terms of very sparse matrices. "MN" (MacKayNeal) codes are recently invented, and "Gallager codes" were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties. The decoding of both codes can be tackled with a practical sumproduct algorithm. We prove that these codes are "very good," in that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit. This result holds not only for the binarysymmetric channel but also for any channel with symmetric stationary ergodic noise. We give experimental results for binarysymmetric channels and Gaussian channels demonstrating that practical performance substantially better than that of standard convolutional and concatenated codes can be achieved; indeed, the performance of Gallager codes is almost as close to the Shannon limit as that of turbo codes.
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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Cited by 800 (26 self)
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The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide varietyof algorithms — among them sumproduct, cluster variational methods, expectationpropagation, mean field methods, maxproduct and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in largescale statistical models.
The Berlekamp algorithm
, 2009
"... Integer factorization is a Hard Problem. Some cryptosystems, such as RSA, are in fact designed around the difficulty of integer factorization. For polynomials with coefficients in the finite field Fq, on the other hand, we can use the Berlekamp algorithm to factor polynomials of high degree in reaso ..."
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Cited by 1 (0 self)
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Integer factorization is a Hard Problem. Some cryptosystems, such as RSA, are in fact designed around the difficulty of integer factorization. For polynomials with coefficients in the finite field Fq, on the other hand, we can use the Berlekamp algorithm to factor polynomials of high degree
The BerlekampMassey Algorithm revisited
"... We propose a slight modification of the BerlekampMassey Algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence. Such a modification enables to explain it in a simpler way and to adapt it to lazy evaluation. ..."
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Cited by 4 (0 self)
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We propose a slight modification of the BerlekampMassey Algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence. Such a modification enables to explain it in a simpler way and to adapt it to lazy evaluation.
1 BerlekampMassey Algorithm
"... The main purpose of this paper is to acquaint the reader with the BerlekampMassey algorithm, it's proof and some of it's applications. In these last six weeks, I have studied many dierent topics ranging from rst year Calculus reviews to second and third year graduate school topics. I chos ..."
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The main purpose of this paper is to acquaint the reader with the BerlekampMassey algorithm, it's proof and some of it's applications. In these last six weeks, I have studied many dierent topics ranging from rst year Calculus reviews to second and third year graduate school topics. I
On the Hardness of Decoding the Gale–Berlekamp Code
, 2007
"... The Gale–Berlekamp (in short, GB) code is the dual code of the binary product code in which the horizontal and vertical constituent codes are both the parity code. It is shown that the problem of deciding whether there is a codeword of the GB code within a prescribed distance from a given received w ..."
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Cited by 9 (0 self)
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The Gale–Berlekamp (in short, GB) code is the dual code of the binary product code in which the horizontal and vertical constituent codes are both the parity code. It is shown that the problem of deciding whether there is a codeword of the GB code within a prescribed distance from a given received
Improved Decoding of ReedSolomon and AlgebraicGeometry Codes
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1999
"... Given an errorcorrecting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding ReedSolomon codes ..."
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Cited by 343 (42 self)
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Given an errorcorrecting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding ReedSolomon codes. The list decoding problem for ReedSolomon codes reduces to the following "curvefitting" problem over a field F : Given n points f(x i :y i )g i=1 , x i
The Solution to Berlekamp’s Switching Game *
"... Berlekamp’s game consists of a 10 × 10 array of lightbulbs, with 100 switches at the back, one for each bulb, and 20 switches at the front that can complement any row or column of bulbs. For any initial set S of bulbs turned on using the back switches, let f (S) be the minimal number of lights that ..."
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Cited by 6 (0 self)
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Berlekamp’s game consists of a 10 × 10 array of lightbulbs, with 100 switches at the back, one for each bulb, and 20 switches at the front that can complement any row or column of bulbs. For any initial set S of bulbs turned on using the back switches, let f (S) be the minimal number of lights
Parallelizing the Big Prime Berlekamp Algorithm with Distributed Maple
, 2001
"... We report on a parallel implementation of the Big Prime Berlekamp algorithm with Distributed Maple. This implementation is based on three parallel subalgorithms with dierent parallelization strategies and dynamic behaviors. The experimental results on a cluster and a multiprocessor system yield supe ..."
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We report on a parallel implementation of the Big Prime Berlekamp algorithm with Distributed Maple. This implementation is based on three parallel subalgorithms with dierent parallelization strategies and dynamic behaviors. The experimental results on a cluster and a multiprocessor system yield
Results 1  10
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