Proximity Gaps for Reed–Solomon Codes

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. Concretely, if $\delta$ is smaller than half the minimal distance of an RS code $v\subset \mathbb{F}_{q}^{n}$, every affine space is either entirely $\delta$-close to the code, or alternatively at most an ($n/q$)-fraction of it is $\delta$-close to the code. 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.