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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Posted Content
Re-Encoding Techniques for Interpolation-Based Decoding of Reed-Solomon Codes
TL;DR: In this paper, interpolation-based decoding of Reed-Solomon codes using the Guruswami-Sudan algorithm (GSA) is considered and the effects of two modification techniques for received vectors, i.e., the reencoding map and the newly introduced periodicity projection, are investigated.
Book ChapterDOI
Protecting Intellectual Property by Guessing Secrets
Marcel Fernandez,Miguel Soriano +1 more
TL;DR: This paper presents a solution to the k = 2 guessing secrets problem consisting in an error correcting code equipped with a tracing algorithm that efficiently recovers the secrets and shows how with a slight modification in the tracing algorithm this approach provides a solutions to the collusion secure fingerprinting problem.
Journal ArticleDOI
Decoding Variants of Reed-Muller Codes over Finite Grids
TL;DR: This paper presents a deterministic algorithm for the unique decoding problem for polynomials of bounded total degree over a general grid S1 Sm.
Journal ArticleDOI
Burst List Decoding of Interleaved Reed–Solomon Codes
Tom Kolan,Ron M. Roth +1 more
TL;DR: It is shown that interleaved Reed-Solomon codes can be list-decoded for burst errors while attaining the generalized Reiger bound for list decoding and a respective decoding algorithm is presented that is (significantly) more efficient than a burst list decoder for a noninterleaving Reed- Solomon code with comparable parameters.
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Collision-Resilient Symbol Based Extension of Query Tree Protocol for Fast RFID Tag Identification
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TL;DR: This work proposes a robust and efficient tag collision recovery scheme using direct sequence spreading modulation and experimented with two collision resilient symbols: orthogonal (Hadamard) and BIBD (balanced incomplete block design) code.
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.
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.