Improved List Size for Folded Reed-Solomon 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.