On one-sided testing affine subspaces

Abstract

We study the query complexity of one-sided ϵ-testing the class of Boolean functions $f:F^n\to \{0,1\}$ that describe affine subspaces and Boolean functions that describe axis-parallel affine subspaces, where $F$ is any finite field. We give polynomial-time ϵ-testers that ask $\tilde{O}(1/ϵ)$ queries. This improves the query complexity $\tilde{O}\left(\right|F|/ϵ)$ in [14]. The almost optimality of the algorithms follows from the lower bound of $\Omega (1/ϵ)$ for the query complexity proved by Bshouty and Goldreich [3].

We then show that any one-sided ϵ-tester with proximity parameter $ϵ<1/|F{|}^d$ for the class of Boolean functions that describe $(n-d)$-dimensional affine subspaces and Boolean functions that describe axis-parallel $(n-d)$-dimensional affine subspaces must make at least $\Omega (1/ϵ+|F{|}^{d-1}\log n)$ and $\Omega (1/ϵ+|F{|}^{d-1}n)$ queries, respectively. This improves the lower bound $\Omega (\log n/\log \log n)$ that is proved in [14] for $F=\mathrm{GF}\left(2\right)$. We also give testers for those classes with query complexity that almost match the lower bounds.¹