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

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 $\rho<1-R-\varepsilon$ for $q=\Theta(n)$, as well as near-capacity bounds over constant alphabets with improved alphabet-size requirements.