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
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Book ChapterDOI
Secure Network Coding: Bounds and Algorithms for Secret and Reliable Communications
Sidharth Jaggi,Michael Langberg +1 more
TL;DR: In this paper, the authors address the task of multicast communication using network coding in the presence of passive eavesdroppers and active jammers, and show that high-rate private and reliable communication via schemes, that are both computationally efficient and distributed, is possible in the settings under study.
Journal Article
Explicit List-Decodable Codes with Optimal Rate for Computationally Bounded Channels.
Ronen Shaltiel,Jad Silbak +1 more
TL;DR: Guruswami et al. as discussed by the authors showed that the existence of poly-time computable pseudo-random generators for small-circuit channels with high probability over the random variable S when attempting to decode.
Posted Content
Using the Trace Operator to repair the Polynomial Reconstruction based Cryptosystem presented at Eurocrypt 2003.
TL;DR: A modi cation of the Augot-Finiasz cryptosystem is presented which appears to resist to an attack enabling an attacker to decrypt any intercepted ciphertext e ciently.
Proceedings ArticleDOI
Exponential error bounds for algebraic soft-decision decoding of Reed Solomon codes
N. Ratnakar,Ralf Koetter +1 more
TL;DR: An algebraic soft decision decoding algorithms of Reed Solomon codes, which assigns suitable weighted-interpolation and factorization algorithms, which perform better in discrete, memoryless channel.
Posted Content
Lattice (List) Decoding Near Minkowski's Inequality
Ethan Mook,Chris Peikert +1 more
TL;DR: A polynomial-time algorithm is proposed that decodes Reed-Solomon codes under error measured in the Euclidean norm using the Koetter-Vardy “soft decision” variant of the Guruswami-Sudan list-decoding algorithm.
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.
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.
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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.
Book
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.
Book
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.