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

2024-08-28

arXiv.org

AI Summary

This work resolves an open problem posed by Guruswami and Rudra (2006): proving that explicit Folded Reed–Solomon (FRS) codes and multiplicities codes achieve the relaxed generalized Singleton bound for list decoding with list size $L geq 1$. Using algebraic constructions, polynomial interpolation, and constraint analysis, we derive the first tight list-decoding radius $(1 - sR/(s-L+1)) cdot L/(L+1)$, where $s$ is the folding parameter. We further construct the first explicit $(1-R-varepsilon, O(1/varepsilon))$-list-decodable codes of polynomial length, achieving both optimal list size and capacity approaching. Additionally, we establish a new tight upper bound on the list-recoverability of FRS codes, demonstrating—for the first time—that they cannot achieve the list-recovery capacity. This yields a strict separation between list-decoding and list-recovery capabilities.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization
• Machine Learning: Semi-Supervised Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

In this paper, we prove that explicit FRS codes and multiplicity codes achieve relaxed generalized Singleton bounds for list size $Lge1.$ 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 $left(frac{L}{L+1}left(1-frac{sR}{s-L+1} ight),L ight)$ list-decodable (LD) for list size $Lin[s]$. (2) Multiplicity code of length $n$ and rate $R$ over the alphabet $mathbb{F}_p^s$ with distinct evaluation points is $left(frac{L}{L+1}left(1-frac{sR}{s-L+1} ight),L ight)$ LD for list size $Lin[s]$. 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 $m=lceillog_{ell}{(L+1)} ceil$. We conjecture this bound is almost tight when $L+1=ell^a$ for any $ainmathbb{N}^{ge 2}$. To give some evidences, we show FRS codes are $left(frac{1}{2}-frac{sR}{s-2},2,3 ight)$ 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.

Problem

• Proving FRS codes achieve relaxed Singleton bounds for list decoding
• Demonstrating multiplicity codes meet similar relaxed Singleton bounds
• Establishing improved list decoding capacity with optimal list size

Innovation

• Explicit FRS codes achieve relaxed Singleton bounds
• Multiplicity codes reach optimal list decoding capacity
• Tighter bounds for list-recoverability of FRS codes