Introduction to finite fields and their applications
Introduction to finite fields and their applications
Decoding of Reed Solomon codes beyond the error-correction bound
Journal of Complexity
List decoding: algorithms and applications
ACM SIGACT News
Limits to list decodability of linear codes
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Expander-Based Constructions of Efficiently Decodable Codes
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Simple extractors for all min-entropies and a new pseudorandom generator
Journal of the ACM (JACM)
List Decoding of Error-Correcting Codes: Winning Thesis of the 2002 ACM Doctoral Dissertation Competition (Lecture Notes in Computer Science)
Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Improved decoding of Reed-Solomon and algebraic-geometry codes
IEEE Transactions on Information Theory
Combinatorial bounds for list decoding
IEEE Transactions on Information Theory
Reed-Solomon codes for correcting phased error bursts
IEEE Transactions on Information Theory
Linear-time encodable/decodable codes with near-optimal rate
IEEE Transactions on Information Theory
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Algorithmic results in list decoding
Foundations and Trends® in Theoretical Computer Science
Concatenated codes can achieve list-decoding capacity
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Fast polynomial factorization and modular composition in small characteristic
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Extractors for Three Uneven-Length Sources
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Better Binary List-Decodable Codes Via Multilevel Concatenation
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Improvements on the Johnson bound for Reed-Solomon codes
Discrete Applied Mathematics
Better binary list decodable codes via multilevel concatenation
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
List decoding and pseudorandom constructions
AAECC'07 Proceedings of the 17th international conference on Applied algebra, algebraic algorithms and error-correcting codes
Efficient list decoding of explicit codes with optimal redundancy
AAECC'07 Proceedings of the 17th international conference on Applied algebra, algebraic algorithms and error-correcting codes
Fast Polynomial Factorization and Modular Composition
SIAM Journal on Computing
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For every 0 0, we present an explicit construction of error-correcting codes of rate R that can be list decoded in polynomial time up to a fraction (1-R-ε) of errors. These codes achieve the "capacity" for decoding from adversarial errors, i.e., achieve the optimal trade-off between rate and error-correction radius. At least theoretically, this meets one of the central challenges in coding theory.Prior to this work, explicit codes achieving capacity were not known for any rate R. In fact, our codes are the first to beat the error-correction radius of 1-√R, that was achieved for Reed-Solomon (RS) codes in [9], for all rates R. (For rates R folded Reed-Solomon codes, which are in fact exactly RS codes, but viewed as a code over a larger alphabet by careful bundling of codeword symbols. Given the ubiquity of RS codes, this is an appealing feature of our result, since the codes we propose are not too far from the ones in actual use.The main insight in our work is that some carefully chosen folded RS codes are "compressed" versions of a related family of Parvaresh-Vardy codes. Further, the decoding of the folded RS codes can be reduced to list decoding the related Parvaresh-Vardy codes. The alphabet size of these folded RS codes is polynomial in the block length. This can be reduced to a constant that depends on the distance ε to capacity using ideas concerning "list recovering" and expander-based codes from [7, 8]. Concatenating the folded RS codes with suitable inner codes also gives us polytime constructible binary codes that can be efficiently list decoded up to the Zyablov bound.