Journal ArticleDOI
Improved decoding of Reed-Solomon and algebraic-geometry codes
Venkatesan Guruswami,Madhu Sudan +1 more
TLDR
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.Abstract:
Given an error-correcting 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 Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following "curve-fitting" problem over a field F: given n points ((x/sub i//spl middot/y/sub i/))/sub i=1//sup n/, x/sub i/, y/sub i//spl isin/F, and a degree parameter k and error parameter e, find all univariate polynomials p of degree at most k such that y/sub i/=p(x/sub i/) for all but at most e values of i/spl isin/(1,...,n). We give an algorithm that solves this problem for e 1/3, where the result yields the first asymptotic improvement in four decades. The algorithm generalizes to solve the list decoding problem for other algebraic codes, specifically alternant codes (a class of codes including BCH codes) and algebraic-geometry codes. In both cases, we obtain a list decoding algorithm that corrects up to n-/spl radic/(n(n-d')) errors, where n is the block length and d' is the designed distance of the code. The improvement for the case of algebraic-geometry codes extends the methods of Shokrollahi and Wasserman (see in Proc. 29th Annu. ACM Symp. Theory of Computing, p.241-48, 1998) and improves upon their bound for every choice of n and d'. We also present some other consequences of our algorithm including a solution to a weighted curve-fitting problem, which may be of use in soft-decision decoding algorithms for Reed-Solomon codes.read more
Citations
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Proceedings ArticleDOI
M-ary hyper phase-shift keying with Reed Solomon encoding and soft decision reliability information
James Caldwell,Clark Robertson +1 more
TL;DR: A RS forward error correction coding in conjunction with M-ary hyper phase-shift keying and soft decision decoding and a system that utilizes RS encoding of the information symbols which are then transmitted with MHPSK is evaluated in terms of probability of bit error and spectral efficiency.
Proceedings ArticleDOI
Improved reliability information for OFDM systems on time-varying frequency-selective fading channels
TL;DR: This paper extends the Distance Metric method to Orthogonal Frequency Division Multiplexing (OFDM) systems on time-varying frequency-selective Rayleigh fading channels and derives the soft reliability information from the frequency domain CSI.
Patent
Decoding method and decoding apparatus
TL;DR: In this paper, a decoding method of performing maximum a posteriori probability (MAP) decoding of selecting one decoded word from one or more decoding word candidates obtained by subjecting a linear code to iterative decoding by comparison of distances between a reception word and each decoding word candidate is presented.
Journal ArticleDOI
List-Decoding Barnes---Wall Lattices
Elena Grigorescu,Chris Peikert +1 more
TL;DR: Tight combinatorial bounds are given on the worst-case list size, showing it to be polynomial in the lattice dimension for any error radius bounded away from the lattices’s minimum distance (in the Euclidean norm).
Posted Content
Cryptanalyzing the Polynomial-Reconstruction based Public-Key System Under Optimal Parameter Choice.
Aggelos Kiayias,Moti Yung +1 more
TL;DR: In this article, the authors showed that the general approach suggested by the cryptosystem works or not, and they also employed the recent powerful list-decoding mechanisms to break it.
References
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Book
The Theory of Error-Correcting Codes
TL;DR: This book presents an introduction to BCH Codes and Finite Fields, and methods for Combining Codes, and discusses self-dual Codes and Invariant Theory, as well as nonlinear Codes, Hadamard Matrices, Designs and the Golay Code.
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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.
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Algebraic Coding Theory
TL;DR: This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory," originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field.
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A Course in Computational Algebraic Number Theory
TL;DR: The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.
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Algebraic Function Fields and Codes
TL;DR: This new edition, published in the series Graduate Texts in Mathematics, has been considerably expanded and contains numerous exercises that help the reader to understand the basic material.