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

Rohan Goyal, Prahladh Harsha, Mrinal Kumar, Ashutosh Shankar

TL;DR

The paper extends near-linear time list-decoding techniques for Folded Reed-Solomon codes and univariate multiplicity codes to the list-recovery setting up to capacity. It achieves this by adapting a lattice-based interpolation framework to construct an interpolation polynomial Q that yields a differential equation satisfied by all close codewords, and by building per-evaluation-point bases that are then merged into a global basis. A near-linear-time differential-equation solver, combined with a pruning step, recovers the full list with high probability and controllable list size. This work broadens the applicability of fast interpolation methods to list recovery, enabling efficient decoding/recovery in settings with per-coordinate uncertainty and multiple candidate derivatives.

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.

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

TL;DR

The paper extends near-linear time list-decoding techniques for Folded Reed-Solomon codes and univariate multiplicity codes to the list-recovery setting up to capacity. It achieves this by adapting a lattice-based interpolation framework to construct an interpolation polynomial Q that yields a differential equation satisfied by all close codewords, and by building per-evaluation-point bases that are then merged into a global basis. A near-linear-time differential-equation solver, combined with a pruning step, recovers the full list with high probability and controllable list size. This work broadens the applicability of fast interpolation methods to list recovery, enabling efficient decoding/recovery in settings with per-coordinate uncertainty and multiple candidate derivatives.

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 -time algorithms for list recovery of Folded Reed-Solomon codes and univariate Multiplicity codes up to capacity, where 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.

Paper Structure

This paper contains 13 sections, 13 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. An overview of the proof
  3. Preliminaries
  4. Notation
  5. Univariate multiplicity codes
  6. Polynomial operators and lattices
  7. Fast construction of differential equation
  8. Construction of basis for a single evaluation point
  9. Combining the bases
  10. Proof of \ref{['thm: Q exists']}
  11. List recovery algorithm
  12. Fast algorithms for solving differential and functional equations
  13. Algorithm and proof of main theorem

Key Result

theorem 1.0

For every $\varepsilon > 0, \ell \in \mathbb{N}$, there is an $s_0 \in \mathbb{N}$ such that for all $s > s_0$, degree parameter $k$, block length $n$ and field $\mathbb{F}$ of characteristic zero or greater than $k$, the following is true. There is a randomized algorithm that when given as input se

Theorems & Definitions (30)

  • theorem 1.0: Main result
  • remark 1.1
  • remark 1.2
  • definition 2.1: Multiplicity codes
  • definition 2.2: Tau operator
  • proposition 2.3
  • proof
  • definition 2.4: polynomial lattice
  • theorem 2.5: polynomial version of Minkowski's theorem
  • theorem 2.6: Alekhnovich2005,GuptaSSV2012
  • ...and 20 more