From List-Decodability to Proximity Gaps

Abstract

Proximity testing for linear codes is a fundamental problem in coding theory with critical applications in cryptographic protocols, blockchain, and distributed storage systems. This work addresses the proximity gaps for linear codes, a crucial aspect for efficiently verifying whether a batch of words is close to a given code. We present a general framework for deriving proximity gaps from the list-decodability properties of the underlying linear code. Our main result show that for any $(p,L)$ -list-decodable code $C\subseteq \mathbb {F}_{q}^{n}$ with minimum relative distance $\Delta _{C}\gt p$ , the probability that a random combination of a batch of $t$ words containing a $\delta $ -far word (for $\delta \le 1-\sqrt {1-p+\varepsilon }$ ) remains $\delta $ -far from $C$ is bounded by $O\left ({{\frac {tL^{2}pn}{q}+\frac {t}{\varepsilon q}}}\right)$ . This result also establishes a form of (mutual) correlated agreement for linear codes, which can be used to strengthen soundness analyses in protocols that rely on proximity testing, thereby reducing query complexity and enabling practical instantiations over smaller finite fields. In particular, we apply our main result to randomly punctured Reed–Solomon codes and folded Reed–Solomon codes—both of which are known to achieve list-decodability up to capacity—and derive linear proximity gaps for these families of codes under the Johnson bound.