Motivated by applications to property testing in the online-erasure model of Kalemaj, Raskhodnikova, and Varma (ITCS 2022 and Theory of Computing 2023), we define and analyze {\em semi-sample-based testers} for Reed-Muller codes. The task in Reed-Muller testing is to determine whether an input function $f: \F^n \to \F$ belongs to the Reed-Muller code or is far from it, using as few point queries to $f$ as possible. Reed-Muller testing is a well-studied task with its roots in both the Property Testing and Probabilistically Checkable Proofs literature. The online-erasure model introduces a twist: after each query made, an adversary may erase up to $t$ points of the input function, potentially thwarting any test in which the queries follow a predictable pattern. Semi-sample-based testers are a hybrid between sample-based testers -- which can only make uniformly random queries to the input function -- and standard testers, which can choose their queries freely. They are designed with the online-erasure model in mind and operate by first choosing some subset $S$ of the domain and then making their queries uniformly at random inside of $S$. We describe semi-sample-based testers for the Reed-Muller code and give an optimal analysis of their soundness. Consequently, we show that semi-sample-based testers are indeed effective in the presence of online erasures, and thereby achieve optimal query complexity for testing the Reed-Muller code in the online-erasure model. This result improves upon prior work of Minzer and Zheng (SODA 2024). As an added bonus, we show that semi-sample-based testers also exist for the lifted affine-invariant codes of Guo, Kopparty, and Sudan (ITCS 2013), thereby providing the first known testers for these codes in the online-erasure model.

翻译：受Kalemaj、Raskhodnikova和Varma（ITCS 2022与Theory of Computing 2023）提出的在线擦除模型中性质测试应用的启发，我们定义并分析了针对Reed-Muller码的{\em 半样本测试器}。Reed-Muller测试的任务是：通过尽可能少的点查询，判定输入函数$f: \F^n \to \F$是否属于Reed-Muller码或与其相距甚远。该测试是性质测试与概率可验证证明领域中的经典课题。在线擦除模型引入了一个挑战：每次查询后，对手可擦除输入函数中最多$t$个点，从而可能阻碍任何遵循可预测模式查询的测试。半样本测试器是样本测试器（仅能对输入函数进行均匀随机查询）与标准测试器（可自由选择查询）的混合体。它们针对在线擦除模型设计，其运行机制为：首先选取域的子集$S$，随后在$S$内进行均匀随机查询。我们描述了一套针对Reed-Muller码的半样本测试器，并对其可靠性进行了最优分析。由此证明，半样本测试器在在线擦除场景中确实有效，从而在在线擦除模型下实现了Reed-Muller码测试的最优查询复杂度。该结果改进了Minzer与Zheng（SODA 2024）的先前工作。此外，我们进一步证明，对于Guo、Kopparty与Sudan（ITCS 2013）提出的提升仿射不变码，同样存在半样本测试器，从而首次为这些码在在线擦除模型中的测试提供了已知方案。