Paper 2020/654

Proximity Gaps for Reed-Solomon Codes

Eli Ben-Sasson and Dan Carmon and Yuval Ishai and Swastik Kopparty and Shubhangi Saraf

Abstract

A collection of sets displays a proximity gap with respect to some property if for every set in the collection, either (i) all members are $\delta$-close to the property in relative Hamming distance or (ii) only a tiny fraction of members are $\delta$-close to the property. In particular, no set in the collection has roughly half of its members $\delta$-close to the property and the others $\delta$-far from it. We show that the collection of affine spaces displays a proximity gap with respect to Reed-Solomon (RS) codes, even over small fields, of size polynomial in the dimension of the code, and the gap applies to any $\delta$ smaller than the Johnson/Guruswami-Sudan list-decoding bound of the RS code. We also show near-optimal gap results, over fields of (at least) linear size in the RS code dimension, for $\delta$ smaller than the unique decoding radius. Finally, we discuss several applications of our proximity gap results to distributed storage, multi-party cryptographic protocols, and concretely efficient proof systems. We prove the proximity gap results by analyzing the execution of classical algebraic decoding algorithms for Reed-Solomon codes (due to Berlekamp-Welch and Guruswami-Sudan) on a formal element of an affine space. This involves working with Reed-Solomon codes whose base field is an (infinite) rational function field. Our proofs are obtained by developing an extension (to function fields) of a strategy of Arora and Sudan for analyzing low-degree tests.