Artin automorphisms, Cyclotomic function fields, and Folded list-decodable codes

TL;DR

This paper introduces a novel approach to constructing folded algebraic-geometric codes using Artin automorphisms in cyclotomic function fields, achieving list decoding capacity with smaller alphabet sizes than previous methods.

Contribution

It develops new folded algebraic-geometric codes based on cyclotomic function fields and Artin automorphisms, improving list decoding efficiency and reducing alphabet size.

Findings

Codes achieve list decoding capacity with polylogarithmic alphabet size.

New insights into algebraic codes and their decoding methods.

Applications to explicit binary codes for list decoding up to the Zyablov radius.

Abstract

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 $M \in \F_q[T]$. 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.