嵌套正交阵列的改进界
Improved Bounds for Nested Orthogonal Arrays
浏览论文内容
中文总结 AI 辅助
本文在群论框架下重新表述Rao界，推导非对称嵌套正交阵列（NOAs）的更紧下界，其在因子水平相等时归约为对称界，且通过构造NOAs验证了新 bounds 的最优性。
中文摘要 AI 辅助
嵌套正交阵列（NOAs）在各类试验设计问题中的应用日益广泛。该领域的核心挑战是推导运行次数的下界，这些下界可作为证明特定阵列不存在的有力判据。对于对称NOAs，Mukerjee、Qian和Wu提出的统计论证能给出相当紧的界；相比之下，Lin、Pang和Chen提出的非对称NOAs的界，是通过递归列删除技术将一般问题归约为强度为2的NOAs得到的，并非最优。因此，改进这些界仍是一个重要的开放问题。本文中，我们重新表述Rao界，并在群论框架下为NOAs建立新的界。利用有限阿贝尔群的特征理论，我们通过群特征得到等价刻画。该框架使我们能给出正交阵列Rao界的新证明，并推导非对称NOAs比Lin等人的界显著更紧的下界。当所有因子水平相等时，我们的界自然归约为Mukerjee、Qian和Wu的对称界，还通过明确构造两个达到这些界的NOAs验证了其最优性。
英文摘要
Nested orthogonal arrays (NOAs) have found increasing application in various experimental design problems. A central challenge in this field is the derivation of lower bounds on the number of runs. These bounds serve as a powerful criterion to prove the nonexistence of specific arrays. For symmetric NOAs, Mukerjee, Qian, and Wu developed statistical arguments that yield fairly tight bounds. By contrast, the bounds for asymmetric NOAs proposed by Lin, Pang and Chen, which are obtained via a recursive column deletion technique that reduces the general problem to NOAs of strength 2, are not optimal. Consequently, improving these bounds remains a significant open problem. In this paper, we reformulate the Rao bound and establish a new bound for NOAs under a group theoretic framework. Using the character theory of finite abelian groups, we obtain an equivalent characterization via group characters. This framework allows us to give a new proof of Rao bound for orthogonal arrays and to derive significantly sharper lower bounds for asymmetric NOAs than those of Lin et al. In the case where all factor levels are equal, our bounds reduce naturally to the symmetric bounds of Mukerjee, Qian, and Wu. We also confirm their optimality by explicitly two constructions of NOAs that achieve these bounds.
