  Linearity tests are randomized algorithms which have oracle access to the
truth table of some function f, and are supposed to distinguish between linear
functions and functions which are far from linear. Linearity tests were first
introduced by (Blum, Luby and Rubenfeld, 1993), 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. 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 (Samorodnitsky and Trevisan 2000) constructed the
Complete Graph Test, and prove that no Hyper Graph Test can perform better than
the Complete Graph Test. Later in (Samorodnitsky and Trevisan 2006) they prove,
among other results, that no non-adaptive linearity test can perform better
than the Complete Graph Test. Their proof uses the algebraic machinery of the
Gowers Norm. A result by (Ben-Sasson, Harsha and Raskhodnikova 2005) allows to
generalize this lower bound also to adaptive linearity tests. We also prove the
same optimal lower bound for adaptive linearity test, but our proof technique
is arguably simpler and more direct than the one used in (Samorodnitsky and
Trevisan 2006). We also study, like (Samorodnitsky and Trevisan 2006), the
behavior of linearity tests on quadratic functions. However, instead of
analyzing the Gowers Norm of certain functions, we provide a more direct
combinatorial proof, studying the behavior of linearity tests on random
quadratic functions...
