arXiv:2611.15128 · v1

Affine Combiners for Reed-Solomon Proximity Gaps

首次提交 2026-11-11 15:13

版本更新 2026-11-11 15:13

雷达观测 2026-11-13 11:11

摘要

Reed-Solomon proximity gaps establish soundness guarantees for protocols that test received words through random combinations. For length-$n$ codes over $\mathbb{F}_q$, classical scalar combiners achieve error $O(n/q)$ in the unique-decoding regime, with larger known bounds near the Johnson radius. We prove smaller error bounds using affine combiners, which combine two received words $f,g$ as $f(x)+(α+βx)g(x)$, where $α,β$ are independent uniform elements of $\mathbb{F}_q$. For output degree bound $d=ρn$, fixed $ρ\in(0,1)$ and $0<η<1-\sqrtρ$, and sufficiently large block length $n$, we prove mutual correlated agreement (MCA) with one coordinate of agreement loss at threshold $A=\lceil(\sqrtρ+η)n\rceil$, with error $O_{ρ,η}\left(\frac{1}{q}+\frac{n^2}{q^2}\right)$. The result holds over arbitrary evaluation domains in every finite field, including prime and binary extension fields. The agreement threshold can approach the Johnson threshold, beyond unique decoding. For $q\ge n^2$, our error is $O_{ρ,η}(1/q)$, an asymptotic factor-$n$ separation from scalar combiners: we prove that their worst-case MCA error is $Ω_{ρ,η}(n/q)$ at the same agreement threshold and with the same one-coordinate loss. We also prove that affine combiners incur worst-case error $Ω_{ρ,η}(n/q)$ when no agreement loss is allowed. Thus one coordinate of agreement loss is necessary for our guarantee; this relaxation of standard MCA reduces the guaranteed common agreement fraction by at most $1/n$. We also extend the result to $k$-variate Reed-Muller codes on subfield grids, obtaining MCA error $O_{ρ,η,k}(q^{-k})$ for fixed $ρ,η,k$, sufficiently large $n$, and $q\ge n^{k+1}$, with agreement loss $n^{1-1/k}$.