Improved Bounds for Testing Juntas

We consider the problem of testing functions for the property of being a k-junta (i.e., of depending on at most kvariables). Fischer, Kindler, Ron, Safra, and Samorodnitsky (J. Comput. Sys. Sci., 2004) showed that $\tilde{O}(k^2)/\epsilon$ queries are sufficient to test k-juntas, and conjectured that this bound is optimal for non-adaptive testing algorithms. Our main result is a non-adaptive algorithm for testing k-juntas with $\tilde{O}(k^{3/2})/\epsilon$ queries. This algorithm disproves the conjecture of Fischer et al. We also show that the query complexity of non-adaptive algorithms for testing juntas has a lower bound of $\min \big(\tilde{\Omega}(k/\epsilon), 2^k/k\big)$, essentially improving on the previous best lower bound of i¾?(k).