An Optimal Tester for k-Linear
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A Boolean function f:{0,1}^n→{0,1} is k-linear if it returns the sum (over the binary field F_2) of 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=0^kj-Linear. We give a non-adaptive distribution-free two-sided ϵ-tester for k-Linear that makes O(klog k+1/ϵ) 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 Ω̃(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 Ω̃(√(k))log n+Ω(1/ϵ) queries.
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