Given Boolean functions \( f, g : \mathbb{F}_2^n \to \{-1,+1\} \), we say they are {\em linearly isomorphic} if there exists \( A \in \mathrm{GL}_n(\mathbb{F}_2) \) such that \( f(x)=g(Ax) \) for all \( x \). We study this problem in the tolerant property testing framework under the known--unknown model, where \( g \) is given explicitly and \( f \) is accessible only via oracle queries, meaning the algorithm may adaptively request the value of \( f(x) \) for inputs \( x \in \mathbb{F}_2^n \) of its choice. Given parameters \( ε\ge 0 \) and \( ω>0 \), the goal is to distinguish whether there exists \( A \in \mathrm{GL}_n(\mathbb{F}_{2})\) such that the normalized Hamming distance between \( f \) and \( g(Ax) \) is at most \( ε\), or whether for every \( A \in \mathrm{GL}_n(\mathbb{F}_2) \) the distance is at least \( ε+ω\). Our main result is a tolerant tester making \( \widetilde{O} \left( \left( m/ω\right)^4 \right) \) queries to \( f \), where \( m \) is an upper bound on the spectral norm of \( g \), improving the previous \( \widetilde{O} \left( \left( m/ω\right)^{24} \right) \) bound of Wimmer and Yoshida. We complement this with a nearly matching lower bound of \( Ω(m^2) \) for constant \( ω\) (for example, \( ω=1/4 \)), improving the prior \( Ω(\log m) \) lower bound of Grigorescu, Wimmer and Xie. A key technical ingredient on the algorithmic side is a query-efficient local list corrector. For the lower bound, we give a reduction from communication complexity using a novel subclass of Maiorana--McFarland functions from symmetric-key cryptography.

翻译：给定布尔函数 \( f, g : \mathbb{F}_2^n \to \{-1,+1\} \)，若存在 \( A \in \mathrm{GL}_n(\mathbb{F}_2) \) 使得对所有 \( x \) 有 \( f(x)=g(Ax) \)，则称它们是{\em 线性同构的}。我们在已知-未知模型下的容忍属性测试框架中研究此问题，其中 \( g \) 被显式给出，而 \( f \) 仅能通过预言查询访问，即算法可以自适应地请求其选择的输入 \( x \in \mathbb{F}_2^n \) 对应的 \( f(x) \) 值。给定参数 \( ε\ge 0 \) 和 \( ω>0 \)，目标是区分是否存在 \( A \in \mathrm{GL}_n(\mathbb{F}_{2})\) 使得 \( f \) 与 \( g(Ax) \) 之间的归一化汉明距离至多为 \( ε\)，或者是否对每个 \( A \in \mathrm{GL}_n(\mathbb{F}_2) \) 该距离至少为 \( ε+ω\)。我们的主要结果是构造了一个容忍测试器，其对 \( f \) 的查询复杂度为 \( \widetilde{O} \left( \left( m/ω\right)^4 \right) \)，其中 \( m \) 是 \( g \) 的谱范数上界，这改进了 Wimmer 和 Yoshida 先前 \( \widetilde{O} \left( \left( m/ω\right)^{24} \right) \) 的界。我们进一步补充了一个在常数 \( ω\)（例如 \( ω=1/4 \)）下近乎匹配的 \( Ω(m^2) \) 下界，改进了 Grigorescu、Wimmer 和 Xie 先前 \( Ω(\log m) \) 的下界。算法侧的一个关键技术要素是一个查询高效的局部列表校正器。对于下界，我们利用对称密钥密码学中 Maiorana--McFarland 函数的一个新颖子类，通过通信复杂性给出了一个归约。