A Boolean function $f:\{0,1\}^n\to \{0,1\}$ 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 $\cup_{j=0}^kj$-Linear. We give a non-adaptive distribution-free two-sided $\epsilon$-tester for $k$-Linear that makes $$O\left(k\log k+\frac{1}{\epsilon}\right)$$ queries. This matches the lower bound known from the literature. We then give a non-adaptive distribution-free one-sided $\epsilon$-tester for $k$-Linear$^*$ that makes the same number of queries and show that any non-adaptive uniform-distribution one-sided $\epsilon$-tester for $k$-Linear must make at least $ \tilde\Omega(k)\log n+\Omega(1/\epsilon)$ queries. The latter bound, almost matches the upper bound $O(k\log n+1/\epsilon)$ known from the literature. We then show that any adaptive uniform-distribution one-sided $\epsilon$-tester for $k$-Linear must make at least $\tilde\Omega(\sqrt{k})\log n+\Omega(1/\epsilon)$ queries.

中文翻译：

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$k$-Linear 的最佳测试器

布尔函数 $f:\{0,1\}^n\to \{0,1\}$ 是 $k$-linear 如果它返回 $k$ 坐标的总和（在二进制字段 $F_2$ 上）的输入。在本文中，我们研究了类 $k$-Linear（所有 $k$-linear 函数的类）和 $k$-Linear$^*$（类 $\cup_{j=0}^）的属性测试kj$-线性。我们为 $k$-Linear 提供了一个非自适应无分布的双面 $\epsilon$-tester，它使得 $$O\left(k\log k+\frac{1}{\epsilon}\right)$$查询。这与文献中已知的下限相匹配。然后，我们为 $k$-Linear$^*$ 给出一个非自适应分布的单边 $\epsilon$-tester，它进行相同数量的查询，并表明任何非自适应均匀分布单边 $ $k$-Linear 的 \epsilon$-tester 必须至少进行 $ \tilde\Omega(k)\log n+\Omega(1/\epsilon)$ 查询。后者绑定，几乎与文献中已知的上限 $O(k\log n+1/\epsilon)$ 匹配。然后，我们证明了 $k$-Linear 的任何自适应均匀分布单边 $\epsilon$-tester 必须至少使 $\tilde\Omega(\sqrt{k})\log n+\Omega(1/\epsilon )$ 查询。