On one-sided testing affine subspaces

We study the query complexity of one-sided epsilon-testing the class of Boolean functions f : F-n -> {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 epsilon-testers that ask (O) over tilde (1/epsilon) queries. This improves the query complexity (O) over tilde(|F|/epsilon) in [14]. The almost optimality of the algorithms follows from the lower bound of Omega(1/epsilon) for the query complexity proved by Bshouty and Goldreich [3]. We then show that any one-sided epsilon-tester with proximity parameter epsilon < 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/epsilon+|F|(d-1)log n) and Omega(1/epsilon + |F|(d-1)n) queries, respectively. This improves the lower bound Omega(log n/log log n) that is proved in [14] for F = GF(2). We also give testers for those classes with query complexity that almost match the lower bounds.(1)