Proceedings ArticleDOI
Linear Diophantine equations over polynomials and soft decoding of Reed-Solomon codes
Michael Alekhnovich
- pp 439-448
TLDR
This paper gives another fast algorithm for the soft decoding of Reed-Solomon codes different from the procedure proposed by Feng, which works in time (w/r) O(1)nlog2n, where r is the rate of the code, and w is the maximal weight assigned to a vertical line.Abstract:
We generalize the classical Knuth-Schonhage algorithm computing GCD of two polynomials for solving arbitrary linear Diophantine systems over polynomials in time, quasi-linear in the maximal degree As an application, we consider the following weighted curve fitting problem: given a set of points in the plain, find an algebraic curve (satisfying certain degree conditions) that goes through each point the prescribed number of times The main motivation for this problem comes from coding theory, namely it is ultimately related to the list decoding of Reed-Solomon codes We present a new fast algorithm for the weighted curve fitting problem, based on the explicit construction of Groebner basis This gives another fast algorithm for soft-decoding of Reed-Solomon codes different from the procedure proposed by Feng (1999), which works in time (w/r)/sup O(1)/ n log/sup 2/ n loglogn, where r is the rate of the code, and w is the maximal weight assigned to a vertical lineread more
Citations
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Proceedings ArticleDOI
Efficiently decodable non-adaptive group testing
TL;DR: This paper presents a randomness efficient construction of list disjunct matrices, expanders, dispersers and disjunction matrices that can be decoded in time poly(i-d), and achieves an efficient decoding time and matches the best known (d2 log n) bound on the number of tests.
Proceedings ArticleDOI
List-decoding using the XOR lemma
TL;DR: It is shown how to reduce advice in Impagliazzo's proof of the Direct Product Lemma for pairwise independent inputs, which leads to error-correcting codes with O(n/sup 2/) encoding length, 0/sup /spl tilde//(n) encoding time, and probabilistic 0/Sup /splTilde// (n) list-decoding time.
Proceedings ArticleDOI
Linear time encodable and list decodable codes
Venkatesan Guruswami,Piotr Indyk +1 more
TL;DR: The first construction of error-correcting codes which can be (list) decoded from a noise fraction arbitrarily close to 1 in linear time from a fraction (1-ε) of errors for arbitrary ε > 0 is presented.
Proceedings ArticleDOI
An Interpolation Algorithm using Gröbner Bases for Soft-Decision Decoding of Reed-Solomon Codes
TL;DR: An efficient algorithm that solves the minimal polynomial of the ideal of interpolating polynomials with respect to a certain monomial order is presented based on the theory of Grobner bases of modules.
Posted Content
The Re-Encoding Transformation in Algebraic List-Decoding of Reed-Solomon Codes
TL;DR: In this article, a technique based on re-encoding and coordinate transformation is proposed to reduce the complexity of the bivariate interpolation procedure. But the technique is not suitable for soft-decoding.
References
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Book
The Design and Analysis of Computer Algorithms
Alfred V. Aho,John E. Hopcroft +1 more
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
Journal ArticleDOI
Improved decoding of Reed-Solomon and algebraic-geometry codes
Venkatesan Guruswami,Madhu Sudan +1 more
TL;DR: An improved list decoding algorithm for decoding Reed-Solomon codes and alternant codes and algebraic-geometry codes is presented and a solution to a weighted curve-fitting problem is presented, which may be of use in soft-decision decoding algorithms for Reed- Solomon codes.
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Algebraic Complexity Theory
TL;DR: This is the first book to present an up-to-date and self-contained account of Algebraic Complexity Theory that is both comprehensive and unified.
Journal ArticleDOI
Decoding of Reed Solomon Codes beyond the Error-Correction Bound
TL;DR: To the best of the knowledge, this is the first efficient (i.e., polynomial time bounded) algorithm which provides error recovery capability beyond the error-correction bound of a code for any efficient code.
Journal ArticleDOI
Algebraic soft-decision decoding of Reed-Solomon codes
R. Koetter,Alexander Vardy +1 more
TL;DR: A polynomial-time soft-decision decoding algorithm for Reed-Solomon codes is developed and it is shown that the asymptotic performance can be approached as closely as desired with a list size that does not depend on the length of the code.
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