On Proximity Gaps for One-Point Algebraic Geometric Codes

Proximity gaps and correlated agreement underpin the soundness of interactive oracle proofs. We establish these properties for general one-point algebraic geometry (AG) codes, extending corresponding results for Reed-Solomon codes. For codes of length n over Fq with designed distance $n-m$, write $ρ=m/n$ and fix ρ∈(0,1). For polynomial curves of degree at most and any 0<ε<1, we prove correlated agreement with multiplicative slack ε and soundness O((+1)/(ε q)) whenever e≤(1-ε)(n-m)/2, with no additional genus penalty. This gives asymptotically good AG code families from Garcia--Stichtenoth tower of unbounded length over a fixed field with soundness bounds independent of n, beyond the RS restriction n≤ q. We also establish mutual correlated agreement up to radius (1-√ρ-η)n, with soundness O( n/(η³q)) for 0<η<1-√ρ. Our half designed distance proof combines auxiliary divisors, simultaneous Berlekamp-Welch interpolation, and error-incidence counting to obtain a common polynomial representation without a Polishchuk-Spielman divisibility lemma. Finally, two counterexamples on the Garcia-Stichtenoth tower show limitations beyond half the minimum distance, at radii below the Johnson bound and with positive slack. The first requires a positive constant soundness lower bound independent of q and n, precluding a uniform $O(1/q)$ guarantee throughout the Johnson regime. The second fixes the field and lets the relative gap above half distance tend to zero while ruling out correlated agreement for every soundness parameter $σ<1$, with positive limiting rate and relative error radius.