Decoding of Reed Solomon codes beyond the error-correction bound
Journal of Complexity
Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Subspace polynomials and limits to list decoding of Reed-Solomon codes
IEEE Transactions on Information Theory
List decoding subspace codes from insertions and deletions
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Folded codes from function field towers and improved optimal rate list decoding
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Improved decoding of Reed-Solomon and algebraic-geometry codes
IEEE Transactions on Information Theory
Linear authentication codes: bounds and constructions
IEEE Transactions on Information Theory
Limits to List Decoding Reed–Solomon Codes
IEEE Transactions on Information Theory
Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy
IEEE Transactions on Information Theory
Coding for Errors and Erasures in Random Network Coding
IEEE Transactions on Information Theory
A Rank-Metric Approach to Error Control in Random Network Coding
IEEE Transactions on Information Theory
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We consider Reed-Solomon (RS) codes whose evaluation points belong to a subfield, and give a linear-algebraic list decoding algorithm that can correct a fraction of errors approaching the code distance, while pinning down the candidate messages to a well-structured affine space of dimension a constant factor smaller than the code dimension. By pre-coding the message polynomials into a subspace-evasive set, we get a Monte Carlo construction of a subcode of Reed-Solomon codes that can be list decoded from a fraction (1-R-ε) of errors in polynomial time (for any fixed ε 0) with a list size of O(1/ε). Our methods extend to algebraic-geometric (AG) codes, leading to a similar claim over constant-sized alphabets. This matches parameters of recent results based on folded variants of RS and AG codes. but our construction here gives subcodes of Reed-Solomon and AG codes themselves (albeit with restrictions on the evaluation points). Further, the underlying algebraic idea also extends nicely to Gabidulin's construction of rank-metric codes based on linearized polynomials. This gives the first construction of positive rate rank-metric codes list decodable beyond half the distance, and in fact gives codes of rate R list decodable up to the optimal (1-R-ε) fraction of rank errors. A similar claim holds for the closely related subspace codes studied by Koetter and Kschischang. We introduce a new notion called subspace designs as another way to pre-code messages and prune the subspace of candidate solutions. Using these, we also get a deterministic construction of a polynomial time list decodable subcode of RS codes. By using a cascade of several subspace designs, we extend our approach to AG codes, which gives the first deterministic construction of an algebraic code family of rate R with efficient list decoding from 1-R-ε fraction of errors over an alphabet of constant size (that depends only on ε). The list size bound is almost a constant (governed by log* (block length)), and the code can be constructed in quasi-polynomial time.