Tight lower bounds for adaptive linearity tests

Linearity tests are randomized algorithms which have oracle access to the truth table of some function f , which are supposed to distinguish between linear functions and functions which are far from linear. Linearity tests were first introduced by Blum, Luby and Rubenfeld in [BLR93], and were later used in the PCP theorem among other applications. The quality of a linearity test is described by its correctness c the probability it accepts linear functions, its soundness s the probability it accepts functions far from linear, and its query complexity q the number of queries it makes. The BLR test had q = 3 and s = 1/2. Linearity tests were studied in order to decrease the soundness of linearity tests, while keeping the query complexity small (for one reason, to improve PCP constructions). Samorodnitsky and Trevisan constructed in [ST00] the Complete Graph Test, which for every k ∈ N has q = ( k 2 ) + k and s = 2−( k 2 ). They prove that no Hyper Graph Test can perform better than the Complete Graph Test. Later in [ST06] they prove, among other results, that no non-adaptive linearity test can perform better than the Complete Graph Test. We generalize their result for adaptive tests, and prove that the Complete Graph Test is optimal even against adaptive linearity tests. Our lower bound is actually proven in a more general setting, considering the Average Query Complexity of a linearity test. Our proof technique is somewhat different from the one used in [ST06]. In both cases the behavior of linearity tests against quadratic functions are considered, but while [ST06] uses algebraic analysis of the Gowers Norm of certain functions, we use a more direct combinatorial approach, which allows us to also handle the case of adaptive linearity tests.