An optimal tester for k-linear

A Boolean function f:{0,1}n→{0,1} is k-linear if it returns the sum (over the binary field F2) of exactly k coordinates of the input. In this paper, we study property testing of the classes k-Linear, the class of all k-linear functions, and k-Linear⁎, the class ∪j=0kj-Linear. We give a non-adaptive distribution-free two-sided ϵ-tester for k-Linear that makesO(klog⁡k+1ϵ) queries. This matches the lower bound known from the literature. We then give a non-adaptive distribution-free one-sided ϵ-tester for k-Linear⁎ that makes the same number of queries and show that any non-adaptive uniform-distribution one-sided ϵ-tester for k-Linear must make at least Ω˜(k)log⁡n+Ω(1/ϵ) queries. The latter bound almost matches the upper bound O(klog⁡n+1/ϵ) known from the literature. We then show that any adaptive uniform-distribution one-sided ϵ-tester for k-Linear must make at least Ω˜(k)log⁡n+Ω(1/ϵ) queries.