Artin automorphisms, cyclotomic function fields, and folded list-decodable codes

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 conducive to list decoding 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 le...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Venkatesan Guruswami
• Keywords: Algorithms, Folded Reed-solomon Codes, List Decoding Algorithm, List Decoding Capacity, STOC 2009