Paper 2024/1810

Linear Proximity Gap for Linear Codes within the 1.5 Johnson Bound

Abstract

We establish a linear proximity gap for linear codes within the one-and-a-half Johnson bound. Specifically, we investigate the proximity gap for linear codes, revealing that any affine subspace is either entirely $\delta$-close to a linear code or nearly all its members are $\delta$-far from it. When $\delta$ is within the one-and-a-half Johnson bound, we prove an upper bound on the number of members (in the affine subspace) that are $\delta$-close to the linear code for the latter case. Our bound is linear in the length of codewords. In comparison, Ben-Sasson, Carmon, Ishai, Kopparty and Saraf [FOCS 2020] work on Reed-Solomon codes and prove a linear bound when $\delta$ is within the unique decoding bound and a quadratic bound when $\delta$ is within the Johnson bound. Note that when the minimum relative distance of the linear code is bigger than 0.77, the one-and-a-half Johnson bound is better than the unique decoding bound. Proximity gaps for linear codes have implications in various code-based protocols. In many cases, a stronger property than individual distance—known as correlated agreement—is required, i.e., functions in the affine subspace are not only $\delta$-close to a linear code but also agree on the same evaluation domain. Our results support this stronger property. Furthermore, mutual correlated agreement, the further strengthening property, is also supported.