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
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
On representations of algebraic-geometry codes
IEEE Transactions on Information Theory
Transitive and self-dual codes attaining the Tsfasman-Vla˘dut$80-Zink bound
IEEE Transactions on Information Theory
Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy
IEEE Transactions on Information Theory
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Algebraic codes that achieve list decoding capacity were recently constructed by a careful "folding" of the Reed-Solomon code. The "low-degree" nature of this folding operation was crucial to the list decoding algorithm. We show how such folding schemes arise out of the Artin-Frobenius automorphism at primes in Galois extensions. Using this approach, we construct new folded algebraic-geometric codes for list decoding based on cyclotomic function fields with a cyclic Galois group. Such function fields are obtained by adjoining torsion points of the Carlitz action of an irreducible M ∈ Fq[T]. The Reed-Solomon case corresponds to the simplest such extension (corresponding to the case M=T). In the general case, we need to descend to the fixed field of a suitable Galois subgroup in order to ensure the existence of many degree one places that can be used for encoding. Our methods shed new light on algebraic codes and their list decoding, and lead to new codes achieving list decoding capacity. Quantitatively, these codes provide list decoding (and list recovery/soft decoding) guarantees similar to folded Reed-Solomon codes but with an alphabet size that is only polylogarithmic in the block length. In comparison, for folded RS codes, the alphabet size is a large polynomial in the block length. This has applications to fully explicit (with no brute-force search) binary concatenated codes for list decoding up to the Zyablov radius.