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

TL;DR

This work studies how local information on low-dimensional affine subspaces dictates global proximity to explicit algebraic codes. It develops a line-wise framework that combines line proximity gaps, line correlated agreement, subspace designs, pruning, and line stitching to achieve proximity gaps up to capacity for Folded Reed--Solomon codes. By instantiating the framework with Folded-Wronskian subspace-design properties and capacity-list decoding with pruning, the authors obtain capacity-level line and affine proximity gaps with explicit parameters $\delta=1-R-\eta$ and $\varepsilon$ bounds that scale with $q$, $n$, and $\eta$. A key contribution is the line stitching technique, which replaces small-list behavior with a structural alignment along a single code line, and a reduction that lifts line results to affine subspaces, yielding affine proximity gaps for FRS codes. The results advance the local-to-global rigidity program for capacity-achieving codes and provide tools potentially applicable to PCP/IPP soundness and related cryptographic protocols.

Abstract

A collection of sets satisfies a $(δ,\varepsilon)$-proximity gap with respect to some property if for every set in the collection, either (i) all members of the set are $δ$-close to the property in (relative) Hamming distance, or (ii) only a small $\varepsilon$-fraction of members are $δ$-close to the property. In a seminal work, Ben-Sasson \textit{et al.}\ showed that the collection of affine subspaces exhibits a $(δ,\varepsilon)$-proximity gap with respect to the property of being Reed--Solomon (RS) codewords with $δ$ 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 $δ$, 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 $(δ,\varepsilon)$-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].

Figures (2)

• Figure 1: After the subspace-design argument clusters all relevant near-codewords into a small affine subspace $H\subseteq C$, we choose a small set of coordinates $S$. The restriction $\pi_S(x)=x|_S$ defines fibers $F_b=\{x:x|_S=b\}$ (dotted vertical lines). Lemma 5.5 ensures $H_S=\{0\}$, so $\pi_S$ is injective on $H$ and every fiber meets $H$ in at most one point. Therefore, when a received word $y_j$ satisfies the pinning event $y_j|_S=f(\alpha_j)|_S$ for some nearby $f(\alpha_j)\in H$, the fiber through $y_j$ singles out that unique element of $H$, turning local agreement on $S$ into global identification.
• Figure 2: Geometric intuition for $(\delta,a,t)$-line stitching: although $f(\alpha)$ may be chosen adversarially among $\delta$-close codewords, many of the chosen codewords must align on a single code-line inside $C$.

Theorems & Definitions (33)

• Theorem 1.1: Main result (Informal)
• Definition 2.1
• Definition 2.2: Folded Reed--Solomon codes
• Lemma 4.1: Folded-Wronskian subspace design
• proof
• Lemma 4.2
• proof
• Definition 4.3: Subspace-design code
• Lemma 4.4: Subspace-design parameters for $\mathrm{FRS}$
• Definition 5.1: Line proximity gap