Linear-algebraic list decoding of folded Reed-Solomon codes

TL;DR

This paper presents a linear-algebraic list decoding algorithm for folded Reed-Solomon codes that simplifies previous methods by removing the need for root-finding over extension fields, enabling quadratic time decoding.

Contribution

It introduces a linear-algebra-based analysis and implementation of list decoding for folded RS codes, improving efficiency and providing insights into constructing codes with better list-decoding properties.

Findings

Decoding can be performed using linear algebra without extension fields.

The algorithm runs in quadratic time, except for a pruning step.

A Monte Carlo construction achieves near-optimal list size bounds.

Abstract

Folded Reed-Solomon codes are an explicit family of codes that achieve the optimal trade-off between rate and error-correction capability: specifically, for any $\eps > 0$, the author and Rudra (2006,08) presented an $n^{O(1/\eps)}$ time algorithm to list decode appropriate folded RS codes of rate $R$ from a fraction $1-R-\eps$ of errors. The algorithm is based on multivariate polynomial interpolation and root-finding over extension fields. It was noted by Vadhan that interpolating a linear polynomial suffices if one settles for a smaller decoding radius (but still enough for a statement of the above form). Here we give a simple linear-algebra based analysis of this variant that eliminates the need for the computationally expensive root-finding step over extension fields (and indeed any mention of extension fields). The entire list decoding algorithm is linear-algebraic, solving one linear system for the interpolation step, and another linear system to find a small subspace of candidate solutions. Except for the step of pruning this subspace, the algorithm can be implemented to run in {\em quadratic} time. The theoretical drawback of folded RS codes are that both the decoding complexity and proven worst-case list-size bound are $n^{\Omega(1/\eps)}$. By combining the above idea with a pseudorandom subset of all polynomials as messages, we get a Monte Carlo construction achieving a list size bound of $O(1/\eps^2)$ which is quite close to the existential $O(1/\eps)$ bound (however, the decoding complexity remains $n^{\Omega(1/\eps)}$). Our work highlights that constructing an explicit {\em subspace-evasive} subset that has small intersection with low-dimensional subspaces could lead to explicit codes with better list-decoding guarantees.