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

TL;DR

This paper develops near-linear time algorithms for list recovery of Folded Reed-Solomon and univariate multiplicity codes, extending previous decoding algorithms to the list recovery setting using lattice-based techniques.

Contribution

It introduces the first near-linear time algorithms for list recovery of these codes, generalizing existing decoding algorithms to the list recovery problem.

Findings

Achieves $ ilde{O}(n)$-time list recovery algorithms for FRS and univariate multiplicity codes.

Builds on lattice-based methods to capture list recovery problems.

Extends the capacity of codes for practical list recovery applications.

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 $\tilde{O}(n)$-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.