Improved List Size for Folded Reed-Solomon Codes

TL;DR

This paper presents explicit folded Reed-Solomon codes with improved list decoding radius and smaller list sizes, advancing the theoretical limits of error correction in coding theory.

Contribution

It introduces new explicit FRS codes with enhanced list decoding capabilities and smaller list sizes, using a novel combinatorial approach and folded Wronskian determinants.

Findings

Achieves list decoding up to radius 1-R- with list size (1/^2)

Provides a general construction for list decoding near k/(k+1)(1-R) with list size ((k-1)^2+1)

Introduces a new combinatorial viewpoint and sharper bounds for FRS codes

Abstract

Folded Reed-Solomon (FRS) codes are variants of Reed-Solomon codes, known for their optimal list decoding radius. We show explicit FRS codes with rate $R$ that can be list decoded up to radius $1-R-\epsilon$ with lists of size $\mathcal{O}(1/ \epsilon^2)$. This improves the best known list size among explicit list decoding capacity achieving codes. We also show a more general result that for any $k\geq 1$, there are explicit FRS codes with rate $R$ and distance $1-R$ that can be list decoded arbitrarily close to radius $\frac{k}{k+1}(1-R)$ with lists of size $(k-1)^2+1$. Our results are based on a new and simple combinatorial viewpoint of the intersections between Hamming balls and affine subspaces that recovers previously known parameters. We then use folded Wronskian determinants to carry out an inductive proof that yields sharper bounds.