Locality of Curve-Decoding and Improved Proximity Gaps

2026-07-09

AI Summary

This work addresses the inferior nearest-neighbor list-decoding performance of existing random code ensembles—such as random linear codes, Reed–Solomon codes, and LDPC codes—compared to subspace design codes, a gap that widens with increasing degree. For the first time, we directly model curve list-decodability through a Local Coordinate Linearity (LCL) property defined as a constraint on the row space, thereby circumventing the parameter loss inherent in traditional proxy properties. Within this framework, we achieve substantial improvements in the list-decoding radius for multiple random code families, matching the performance of subspace design codes. Moreover, our approach enables black-box transferability: any future enhancement developed for subspace design codes automatically extends to these random ensembles without modification.

Abstract

Proximity gaps are a property of error correcting codes that arise in the study of Interactive Oracle Proofs (IOPs) and Succinct Non-interactive Arguments of Zero Knowledge (SNARKs). Recent work of Goyal and Guruswami has established near-optimal proximity gaps for many families of codes, including subspace design codes, as well as random ensembles like random linear codes, Reed-Solomon codes with random evaluation points, and Gallager's ensemble of LDPC codes (Goyal & Guruswami, 2025). However, the parameters for these latter randomized ensembles are worse than the parameters for subspace design codes, and degrade as the degree ell increases. In this work, we obtain improved proximity gaps for random ensembles of codes, including random linear codes, Reed-Solomon codes with random evaluation points, and Gallager's ensemble. Quantitatively, our results for these random ensembles match the results that Goyal and Guruswami attained for subspace design codes. In fact, our techniques are a black-box transference from subspace design codes: any progress on subspace design codes will automatically lead to analogous progress for these random ensembles. To obtain our results, we extend the Local Coordinate-wise Linear (LCL) property framework developed by Levi, Mosheiff, and Shagrithaya and by Brakensiek, Chen, Dhar, and Zhang to a \textit{row-span constrained} version (Levi, Mosheiff & Shagrithaya, 2025; Brakensiek, Chen, Dhar & Zhang, 2025). This allows us to cast \textit{curve-decodability} -- a property that implies proximity gaps -- directly as a row-span constrained LCL property, and make use of that machinery. In contrast, because curve-decodability is not obviously a vanilla LCL property, prior work had worked with a proxy property instead, leading to the aforementioned parameter losses.

Problem

• proximity gaps
• random ensembles
• error correcting codes
• curve-decodability
• Interactive Oracle Proofs

Innovation

• proximity gaps
• curve-decodability
• row-span constrained LCL
• random ensembles
• error correcting codes