Cyclotomic function fields, Artin–Frobenius automorphisms, and list error correction with optimal rate

Abstract

Algebraic error-correcting codes that achieve the optimal trade-off between rate and fraction of errors corrected (in the model of list decoding) 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 useful for 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 $M\in {\mathbb{F}}_q\left[T\right]$. 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 1 places that can be used for encoding.

Our methods shed new light on algebraic codes and their list decoding, and lead to new codes with optimal trade-off between rate and error correction radius. 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.