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46
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
Decoding Reed Solomon Codes beyond the ErrorCorrection Bound
, 1997
"... We present a randomized algorithm which takes as input n distinct points f(xi; yi)g n i=1 from F \Theta F (where F is a field) and integer parameters t and d and returns a list of all univariate polynomials f over F in the variable x of degree at most d which agree with the given set of points in a ..."
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Cited by 277 (18 self)
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We present a randomized algorithm which takes as input n distinct points f(xi; yi)g n i=1 from F \Theta F (where F is a field) and integer parameters t and d and returns a list of all univariate polynomials f over F in the variable x of degree at most d which agree with the given set of points in at least t places (i.e., yi = f (xi) for at least t values of i), provided t = \Omega (
List Decoding: Algorithms and Applications
 SIGACT News
, 2000
"... Over the years coding theory and complexity theory have benefited from a number of mutually enriching connections. This article focuses on a new connection that has emerged between the two topics in the recent years. This connection is centered around the notion of "listdecoding" for ..."
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Cited by 39 (0 self)
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Over the years coding theory and complexity theory have benefited from a number of mutually enriching connections. This article focuses on a new connection that has emerged between the two topics in the recent years. This connection is centered around the notion of "listdecoding" for errorcorrecting codes. In this survey we describe the listdecoding problem, the algorithms that have been developed, and a diverse collection of applications within complexity theory. 1
Towards Factoring Bivariate Approximate Polynomials
"... A new algorithm is presented for factoring bivariate approximate polynomials over C [x, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the de ..."
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Cited by 25 (0 self)
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A new algorithm is presented for factoring bivariate approximate polynomials over C [x, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the degree of the problem. This method has been implemented in Maple, and has been demonstrated to be efficient and numerically robust.
Decomposition of polytopes and polynomials
 Discrete and Computational Geometry
"... Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons is NPcomplete then present a pseudopolynomial time ..."
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Cited by 23 (6 self)
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Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons is NPcomplete then present a pseudopolynomial time algorithm for decomposing polygons. For higher dimensional polytopes, we give a heuristic algorithm which is based upon projections and uses randomization. Applications of our algorithms include absolute irreducibility testing and factorization of polynomials via their Newton polytopes. 1
Lifting and recombination techniques for absolute factorization
 J. Complexity
, 2007
"... Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic ..."
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Cited by 23 (7 self)
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Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.
On the Hardness of Permanent
, 1999
"... . We prove that if there is a polynomial time algorithm which computes the permanent of a matrix of order n for any inverse polynomial fraction of all inputs, then there is a BPP algorithm computing the permanent for every matrix. It follows that this hypothesis implies P #P = BPP. Our algorithm w ..."
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Cited by 22 (3 self)
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. We prove that if there is a polynomial time algorithm which computes the permanent of a matrix of order n for any inverse polynomial fraction of all inputs, then there is a BPP algorithm computing the permanent for every matrix. It follows that this hypothesis implies P #P = BPP. Our algorithm works over any sufficiently large finite field (polynomially larger than the inverse of the assumed success ratio), or any interval of integers of similar range. The assumed algorithm can also be a probabilistic polynomial time algorithm. Our result is essentially the best possible based on any black box assumption of permanent solvers, and is a simultaneous improvement of the results of Gemmell and Sudan [GS92], Feige and Lund [FL92] as well as Cai and Hemachandra [CH91], and Toda (see [ABG90]). 1 Introduction The permanent of an n \Theta n matrix A is defined as per(A) = X oe2Sn n Y i=1 A i;oe(i) ; where Sn is the symmetric group on n letters, i.e., the set of all permutations of f1;...