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

Yeyuan Chen, Zihan Zhang

TL;DR

This work resolves a central open problem by showing explicit folded Reed-Solomon and univariate multiplicity codes achieve LD capacity with the optimal list size $O(1/\varepsilon)$ under a relaxed generalized Singleton bound. It introduces a geometric framework based on geometric agreement hypergraphs and folded Wronskians to bound the list size, and extends the approach to list-recovery, deriving a tighter radius bound that rules out LR-capacity for these explicit codes. The results establish that Folded RS and univariate multiplicity codes are LD-capacity achievers over polynomial alphabets, while also proving intrinsic separations between list-decoding and list-recovery performance. The methods yield a modular, design-theoretic perspective via strong subspace designs, offering potential pathways to broader explicit capacity-achieving code families and connections to seeded condensers.

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 $s=Θ(1/ε^2)$ and $L=O(1/ε)$, our results imply that both FRS codes and multiplicity codes achieve LD capacity $1-R-ε$ with optimal list size $O(1/ε)$. This exponentially improves the previous state of the art $(1/ε)^{O(1/ε)}$ 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 $(1-R-ε,O(1/ε))$ 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 $(1-R-ε, \ell, O(\frac{\ell}ε))$. This implies an intrinsic separation between LD and LR of FRS codes.

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

TL;DR

This work resolves a central open problem by showing explicit folded Reed-Solomon and univariate multiplicity codes achieve LD capacity with the optimal list size under a relaxed generalized Singleton bound. It introduces a geometric framework based on geometric agreement hypergraphs and folded Wronskians to bound the list size, and extends the approach to list-recovery, deriving a tighter radius bound that rules out LR-capacity for these explicit codes. The results establish that Folded RS and univariate multiplicity codes are LD-capacity achievers over polynomial alphabets, while also proving intrinsic separations between list-decoding and list-recovery performance. The methods yield a modular, design-theoretic perspective via strong subspace designs, offering potential pathways to broader explicit capacity-achieving code families and connections to seeded condensers.

Abstract

In this paper, we prove that explicit FRS codes and multiplicity codes achieve relaxed generalized Singleton bounds for list size Specifically, we show the following: (1) FRS code of length and rate over the alphabet with distinct evaluation points is list-decodable (LD) for list size . (2) Multiplicity code of length and rate over the alphabet with distinct evaluation points is LD for list size . Choosing and , our results imply that both FRS codes and multiplicity codes achieve LD capacity with optimal list size . This exponentially improves the previous state of the art 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 LD codes of rate 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 LR where . We conjecture this bound is almost tight when for any . To give some evidences, we show FRS codes are LR, which proves the tightness in the smallest non-trivial case. Our bound refutes the possibility that FRS codes could achieve LR capacity . This implies an intrinsic separation between LD and LR of FRS codes.
Paper Structure (35 sections, 28 theorems, 46 equations, 1 figure)

This paper contains 35 sections, 28 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. List-decoding.
  3. List-decodability of Reed--Solomon codes.
  4. List-decodability of folded Reed--Solomon codes.
  5. List-decodability of univariate multiplicity codes.
  6. List-recovery.
  7. List-recoverability of folded RS and univariate multiplicity codes.
  8. Main results
  9. List-decodability of folded Reed--Solomon codes.
  10. List-decodability of univariate multiplicity codes.
  11. More general results for list-decoding.
  12. List-recoverability of folded Reed–Solomon codes.
  13. Concurrent work.
  14. Notation.
  15. Paper organization.
  16. ...and 20 more sections

Key Result

Theorem 1.3

For any integers $s,n,L\geq 1$, $k\in [n]$, generator $\gamma$ of $\mathbb{F}^{\times}_q$ and appropriate $(\alpha_1,\alpha_2,\dots,\alpha_n)\in\mathbb{F}_q^n$, the code $\mathsf{FRS}_{n,k}^{(s,\gamma)}(\alpha_1,\alpha_2,\dots,\alpha_n)$ over the alphabet $\mathbb{F}_q^s$ is $\left(\frac{L}{L+1}\lef

Figures (1)

  • Figure 1: An example of our construction in \ref{['thm:listrec']} is provided here, with parameters set to $m=4$, $\lfloor\frac{k-1}{s}\rfloor=7$, and $p=9$. For each $i\in[m]$, $f_i(X)$ is a product of distinct factors in the set $\left\{Q_u(X)\right\}_{u\in[p]}$, listed in the $i$-th row of the table. The factors $Q_u$ are arranged in ascending order in $u$. Each $Q_u$ appears exactly $m-1$ times in the table by the definition of the polynomials $f_i$. Since there are $m$ rows and $p=\left\lfloor\frac{m}{m-1}\lfloor\frac{k-1}{s}\rfloor\right\rfloor$ distinct factors $Q_u$, with each factor appearing exactly $m-1$ times, the way we fill these factors in the table guarantees that each row contains no more than $\left\lceil\frac{p(m-1)}{m}\right\rceil \leq\lfloor\frac{k-1}{s}\rfloor$ factors $Q_u$. As each $Q_u$ has degree $s$, it follows that $\deg{f_i}\leq s\lfloor\frac{k-1}{s}\rfloor\leq k-1$ for $i\in[m]$.

Theorems & Definitions (72)

  • Definition 1.2: Appropriate evaluation points
  • Theorem 1.3
  • Corollary 1.4: Informal, see \ref{['cor:exact_fold']}
  • Theorem 1.5
  • Corollary 1.6: Informal, see \ref{['cor:exact_mult']}
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Remark
  • Definition 2.1: Geometric agreement hypergraph based on FRS codes
  • ...and 62 more