Fast list recovery of univariate multiplicity and folded Reed-Solomon codes

2025-11-28

AI Summary

This work investigates list recovery for Folded Reed–Solomon codes and univariate multiplicity codes, aiming to overcome the limitation that existing near-linear-time algorithms apply only to list decoding. Methodologically, it extends lattice basis reduction to the list recovery setting by constructing structured lattices tailored to the univariate polynomial ring, thereby precisely encoding the recovery constraints. Leveraging algebraic structure design and efficient lattice reduction, the algorithm achieves $ ilde{O}(n)$ time complexity (where $n$ is the code length) and attains the list recovery capacity bound under natural parameter regimes. The key contributions are: (i) the first near-linear-time list recovery algorithm for both Folded Reed–Solomon and univariate multiplicity codes—two fundamental families of algebraic codes; and (ii) a novel paradigm for modeling list recovery via structured lattices over polynomial rings.

Technology Category

• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Other Foundations of Machine Learning

Application Category

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

Abstract

A recent work of Goyal, Harsha, Kumar and Shankar gave nearly linear time algorithms for the list decoding of Folded Reed-Solomon codes (FRS) and univariate multiplicity codes up to list decoding capacity in their natural setting of parameters. A curious aspect of this work was that unlike most list decoding algorithms for codes that also naturally extend to the problem of list recovery, the algorithm in the work of Goyal et al. seemed to be crucially tied to the problem of list decoding. In particular, it wasn't clear if their algorithm could be generalized to solve the problem of list recovery FRS and univariate multiplicity codes in near linear time. In this work, we address this question and design $ ilde{O}(n)$-time algorithms for list recovery of Folded Reed-Solomon codes and univariate Multiplicity codes up to capacity, where $n$ is the blocklength of the code. For our proof, we build upon the lattice based ideas crucially used by Goyal et al. with one additional technical ingredient - we show the construction of appropriately structured lattices over the univariate polynomial ring that emph{capture} the list recovery problem for these codes.

Problem

• Develop near-linear time list recovery algorithms
• Extend lattice-based methods to univariate polynomial rings
• Achieve capacity for Folded Reed-Solomon and multiplicity codes

Innovation

• Lattice-based construction for polynomial ring
• Near-linear time list recovery algorithm
• Structured lattices capture list recovery problem