Artin automorphisms, cyclotomic function fields, and folded list-decodable codes
Venkatesan Guruswami · 2009
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.