Proximity Gaps Conjecture Fails Near Capacity over Prime Fields

TL;DR

This paper demonstrates that for specific Reed-Solomon codes, proximity gaps do not exist at rates just below the capacity, particularly at radii decreasing as 1 over log n, over prime fields.

Contribution

It provides a proof that proximity gaps fail near capacity for a family of Reed-Solomon codes, refining understanding of decoding limits over prime fields.

Findings

Proximity gaps fail at radii O(1/ log n) below capacity.

The result applies to a certain family of Reed-Solomon codes.

The proof builds on a sketch by Krachun and Kazanin.

Abstract

In this report we flesh out a sketch by Krachun and Kazanin to prove that for a certain family of Reed-Solomon codes, proximity gaps fail at radii that are $O(1/\log n)$ below the capacity rate of the code, where $n$ is the length of the code.