Explicit Folded Reed-Solomon and Multiplicity Codes Achieve Relaxed Generalized Singleton Bounds

TL;DR

This paper proves that explicit folded Reed-Solomon and multiplicity codes achieve near-optimal list decoding bounds, resolving open problems and providing the first explicit constructions of capacity-achieving list decoding codes with polynomial-sized alphabets.

Contribution

The paper establishes that explicit FRS and multiplicity codes attain relaxed generalized Singleton bounds, achieving list decoding capacity with optimal list size, and resolves longstanding open problems in coding theory.

Findings

FRS and multiplicity codes are list-decodable up to capacity bounds.

Explicit constructions of capacity-achieving list decoding codes with polynomial-sized alphabets.

New bounds on list-recoverability and separation between list decoding and list recoverability.

Abstract

In this paper, we prove that explicit FRS codes and multiplicity codes achieve relaxed generalized Singleton bounds for list size $L\ge1.$ Specifically, we show the following: (1) FRS code of length $n$ and rate $R$ over the alphabet $\mathbb{F}_q^s$ with distinct evaluation points is $\left(\frac{L}{L+1}\left(1-\frac{sR}{s-L+1}\right),L\right)$ list-decodable (LD) for list size $L\in[s]$. (2) Multiplicity code of length $n$ and rate $R$ over the alphabet $\mathbb{F}_p^s$ with distinct evaluation points is $\left(\frac{L}{L+1}\left(1-\frac{sR}{s-L+1}\right),L\right)$ LD for list size $L\in[s]$. Choosing $s=\Theta(1/\epsilon^2)$ and $L=O(1/\epsilon)$, our results imply that both FRS codes and multiplicity codes achieve LD capacity $1-R-\epsilon$ with optimal list size $O(1/\epsilon)$. This exponentially improves the previous state of the art $(1/\epsilon)^{O(1/\epsilon)}$ established by Kopparty et. al. (FOCS 2018) and Tamo (IEEE TIT, 2024). In particular, our results on FRS codes fully resolve a open problem proposed by Guruswami and Rudra (STOC 2006). Furthermore, our results imply the first explicit constructions of $(1-R-\epsilon,O(1/\epsilon))$ LD codes of rate $R$ with poly-sized alphabets. Our method can also be extended to analyze the list-recoverability (LR) of FRS codes. We provide a tighter radius upper bound that FRS codes cannot be $(\frac{L+1-\ell}{L+1}(1-\frac{mR}{m-1})+o(1),\ell, L)$ LR where $m=\lceil\log_{\ell}{(L+1)}\rceil$. We conjecture this bound is almost tight when $L+1=\ell^a$ for any $a\in\mathbb{N}^{\ge 2}$. To give some evidences, we show FRS codes are $\left(\frac{1}{2}-\frac{sR}{s-2},2,3\right)$ LR, which proves the tightness in the smallest non-trivial case. Our bound refutes the possibility that FRS codes could achieve LR capacity $(1-R-\epsilon, \ell, O(\frac{\ell}{\epsilon}))$. This implies an intrinsic separation between LD and LR of FRS codes.