Highly resilient correctors for polynomials
Information Processing Letters
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
Algorithmic results in list decoding
Foundations and Trends® in Theoretical Computer Science
Linear-Algebraic List Decoding of Folded Reed-Solomon Codes
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
Improved decoding of Reed-Solomon and algebraic-geometry codes
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
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
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The classical family of [n, k]q Reed-Solomon codes over a field Fq consist of the evaluations of polynomials f ∈ Fq[X] of degree k at n distinct field elements. In this work, we consider a closely related family of codes, called (order m) derivative codes and defined over fields of large characteristic, which consist of the evaluations of f as well as its first m - 1 formal derivatives at n distinct field elements. For large enough m, we show that these codes can be list-decoded in polynomial time from an error fraction approaching 1 - R, where R = k/(nm) is the rate of the code. This gives an alternate construction to folded Reed-Solomon codes for achieving the optimal trade-off between rate and list error-correction radius. Our decoding algorithm is linear-algebraic, and involves solving a linear system to interpolate a multivariate polynomial, and then solving another structured linear system to retrieve the list of candidate polynomials f. The algorithm for derivative codes offers some advantages compared to a similar one for folded Reed-Solomon codes in terms of efficient unique decoding in the presence of side information.