Improved List Size for Folded Reed-Solomon Codes
Shashank Srivastava
TL;DR
The paper addresses explicit list decoding for folded Reed-Solomon codes, achieving near-capacity performance with much smaller lists. It introduces a bottom-up, combinatorial view of intersections between Hamming balls and affine subspaces, complemented by folded Wronskian determinant techniques to control linear dependencies across folded coordinates. The main results establish an $O(1/\varepsilon^2)$ list size for decoding up to radius $1-R-\varepsilon$, and a general $(k-1)^2+1$ bound for decoding near radius $\frac{k}{k+1}(1-R)$, valid when the folding parameter $m$ is large relative to $k$. These bounds bring explicit capacity-achieving codes closer to optimal list sizes and open avenues for deterministic near-linear-time decoding, with concurrent work confirming optimal list sizes in related settings.
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 $1-R-ε$ with lists of size $\mathcal{O}(1/ ε^2)$. 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.