Optimal rate algebraic list decoding using narrow ray class fields

TL;DR

This paper constructs algebraic-geometric codes using class field theory that are nearly optimal in rate and can be efficiently list decoded up to the Singleton bound over constant-sized alphabets.

Contribution

It introduces a novel construction of folded AG codes via narrow ray class fields, achieving near-optimal list decoding over constant-sized alphabets without concatenation.

Findings

Codes are within $2/(\sqrt{\ell}-1)$ of the Singleton bound.

List decoding up to error fraction $1-R-\eps$ is achievable in polynomial time.

Codes are constructed over constant-sized alphabets with near-optimal list decoding capabilities.

Abstract

We use class field theory, specifically Drinfeld modules of rank 1, to construct a family of asymptotically good algebraic-geometric (AG) codes over fixed alphabets. Over a field of size $\ell^2$, these codes are within $2/(\sqrt{\ell}-1)$ of the Singleton bound. The functions fields underlying these codes are subfields with a cyclic Galois group of the narrow ray class field of certain function fields. The resulting codes are "folded" using a generator of the Galois group. This generalizes earlier work by the first author on folded AG codes based on cyclotomic function fields. Using the Chebotarev density theorem, we argue the abundance of inert places of large degree in our cyclic extension, and use this to devise a linear-algebraic algorithm to list decode these folded codes up to an error fraction approaching $1-R$ where $R$ is the rate. The list decoding can be performed in polynomial time given polynomial amount of pre-processed information about the function field. Our construction yields algebraic codes over constant-sized alphabets that can be list decoded up to the Singleton bound --- specifically, for any desired rate $R \in (0,1)$ and constant $\eps > 0$, we get codes over an alphabet size $(1/\eps)^{O(1/\eps^2)}$ that can be list decoded up to error fraction $1-R-\eps$ confining close-by messages to a subspace with $N^{O(1/\eps^2)}$ elements. Previous results for list decoding up to error-fraction $1-R-\eps$ over constant-sized alphabets were either based on concatenation or involved taking a carefully sampled subcode of algebraic-geometric codes. In contrast, our result shows that these folded algebraic-geometric codes {\em themselves} have the claimed list decoding property.