Optimal Proximity Gap for Folded Reed--Solomon Codes via Subspace Designs

TL;DR

This paper proves that folded Reed--Solomon codes exhibit a proximity gap property up to the optimal capacity regime, extending previous results known for affine subspaces and Reed--Solomon codes.

Contribution

It establishes the existence of a proximity gap for folded Reed--Solomon codes up to the capacity regime, generalizing prior work on affine subspaces and RS codes.

Findings

Folded Reed--Solomon codes exhibit a proximity gap up to the capacity regime.

The framework applies to suitable subspace-design codes.

Supports recent results showing FRS codes' properties similar to random linear codes.

Abstract

A collection of sets satisfies a $(\delta,\varepsilon)$-proximity gap with respect to some property if for every set in the collection, either (i) all members of the set are $\delta$-close to the property in (relative) Hamming distance, or (ii) only a small $\varepsilon$-fraction of members are $\delta$-close to the property. In a seminal work, Ben-Sasson \textit{et al.}\ showed that the collection of affine subspaces exhibits a $(\delta,\varepsilon)$-proximity gap with respect to the property of being Reed--Solomon (RS) codewords with $\delta$ up to the so-called Johnson bound for list decoding. Their technique relies on the Guruswami--Sudan list decoding algorithm for RS codes, which is guaranteed to work in the Johnson bound regime. Folded Reed--Solomon (FRS) codes are known to achieve the optimal list decoding radius $\delta$, a regime known as capacity. Moreover, a rich line of list decoding algorithms was developed for FRS codes. It is then natural to ask if FRS codes can be shown to exhibit an analogous $(\delta,\varepsilon)$-proximity gap, but up to the so-called optimal capacity regime. We answer this question in the affirmative (and the framework naturally applies more generally to suitable subspace-design codes). An additional motivation to understand proximity gaps for FRS codes is the recent results [BCDZ'25] showing that they exhibit properties similar to random linear codes, which were previously shown to be related to properties of RS codes with random evaluation points in [LMS'25], as well as codes over constant-size alphabet based on AEL [JS'25].