Optimal rate list decoding over bounded alphabets using algebraic-geometric codes

TL;DR

This paper introduces new algebraic-geometric code constructions that are efficiently list decodable close to the optimal error fraction, with nearly optimal alphabet size and small list sizes, advancing the theoretical limits of error correction.

Contribution

It presents two novel classes of algebraic-geometric codes with near-optimal list decoding capabilities, using algebraic and combinatorial techniques, and achieves explicit constructions with improved parameters.

Findings

Codes are list-decodable up to a $1-R- ext{epsilon}$ error fraction.

Achieves list sizes bounded by $O(1/ ext{epsilon})$ and $O( ext{log}^{(r)} N)$ for fixed $r$.

Parameters are close to existential bounds in error radius, alphabet size, and list size.

Abstract

We give new constructions of two classes of algebraic code families which are efficiently list decodable with small output list size from a fraction $1-R-\epsilon$ of adversarial errors where $R$ is the rate of the code, for any desired positive constant $\epsilon$. The alphabet size depends only $\epsilon$ and is nearly-optimal. The first class of codes are obtained by folding algebraic-geometric codes using automorphisms of the underlying function field. The list decoding algorithm is based on a linear-algebraic approach, which pins down the candidate messages to a subspace with a nice "periodic" structure. The list is pruned by precoding into a special form of "subspace-evasive" sets, which are constructed pseudorandomly. Instantiating this construction with the Garcia-Stichtenoth function field tower yields codes list-decodable up to a $1-R-\epsilon$ error fraction with list size bounded by $O(1/\epsilon)$, matching the existential bound up to constant factors. The parameters we achieve are thus quite close to the existential bounds in all three aspects: error-correction radius, alphabet size, and list-size. The second class of codes are obtained by restricting evaluation points of an algebraic-geometric code to rational points from a subfield. Once again, the linear-algebraic approach to list decoding to pin down candidate messages to a periodic subspace. We develop an alternate approach based on "subspace designs" to precode messages. Together with the subsequent explicit constructions of subspace designs, this yields a deterministic construction of an algebraic code family of rate $R$ with efficient list decoding from $1-R-\epsilon$ fraction of errors over a constant-sized alphabet. The list size is bounded by a very slowly growing function of the block length $N$; in particular, it is at most $O(\log^{(r)} N)$ (the $r$'th iterated logarithm) for any fixed integer $r$.