AI 中文总结
针对重数至少为3的P-和Q-多项式关联方案的球面嵌入，研究其球面设计强度，将上界从8改进至5，确定了达到上界的例子，并引入了一种多项式方法。
AI 中文摘要
我们证明，对于具有至少3个类且关于Q-幂等元的P-和Q-多项式关联方案的球面嵌入，若其重数至少为3，则该嵌入作为球面设计的强度至多为5。我们还确定了达到该强度上界的例子。我们的结果改进了Suda此前提出的8的上界[《组合设计杂志》19卷（2011）]，并被认为与Lewis[《离散数学》223卷（2000）]和Miklavič[《电子组合学杂志》32卷（2025）]关于直径和度数均至少为3的Q-多项式距离正则图的围长的结果是对偶的。为建立我们的上界，我们引入并讨论了一种多项式方法，该方法通过构造在球面嵌入的每个点处都消失的适当多项式来发挥作用。
英文摘要
We show that the strength as a spherical design of the spherical embedding of a $P$- and $Q$-polynomial association scheme with at least three classes with respect to a $Q$-polynomial idempotent is at most five, provided that the multiplicity is at least three. We also identify the examples that attain this upper bound on the strength. Our result improves on Suda's earlier upper bound of eight [J. Combin. Des. 19 (2011)], and is considered dual to the results of Lewis [Discrete Math. 223 (2000)] and Miklavič [Electron. J. Combin. 32 (2025)] concerning the girth of a $Q$-polynomial distance-regular graph with diameter and valency both at least three. To establish our upper bound, we introduce and discuss a polynomial method that works by constructing an appropriate polynomial that vanishes at every point of the spherical embedding.
Comments26 pages
