A Syndrome-Space Approach to Proximity Gaps and Correlated Agreement for Random Linear Codes

Abstract: Proximity gaps and correlated agreement have become central tools in the analysis of interactive oracle proofs of proximity (IOPPs) and code-based SNARKs. Informally, a proximity-gap statement says that for a structured set of words -- such as a line, an affine space, or a curve -- either all points are close to the code, or most are far from it. Such statements are essential in sampling-based proof systems, where a verifier queries only a few random locations on a structured object but must still obtain a global soundness guarantee. In Reed--Solomon-based proof systems, one would ideally like the proximity parameter to approach the information-theoretic limit $1-R$, since this is the largest possible radius for a rate-$R$ code and directly affects protocol efficiency. While recent work has substantially strengthened the picture for algebraic codes and linked proximity gaps to decoding-related structural properties, it remains unclear whether analogous results for random linear codes can be proved directly, rather than through decoding-theoretic surrogates. In this work, we establish a direct approach to proximity gaps and correlated agreement for random linear codes in the random parity-check-matrix model, without relying on list decoding as the main engine of the proof. Our approach is based on a syndrome-space reformulation together with a witness-based reduction mechanism, and it yields strong results for affine lines, affine spaces, and polynomial curves. It is conceptually different from the existing decoding-driven route for random linear codes, and it also leads to sharper parameters, including the optimal-up-to-$\varepsilon$ large-alphabet radius bound $ρ<1-R-\varepsilon$ for $q=Θ(n)$, as well as near-capacity bounds over constant alphabets with improved alphabet-size requirements.