发表机构
Dover Networks LLC（多弗网络有限责任公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对有限平移平面中的扩扑构造，提出计算算法枚举PG(3,q)正则扩扑中可替换的reguli、链和巢，并改进(q-1)-巢族，同时给出反例推翻t-巢可替换的猜想。
AI 中文摘要
有限平移平面的理论紧密联系于PG(3,q)的扩扑理论，其中扩扑是将PG(3,q)划分为$q^2+1$条两两不相交的线。生成扩扑的传统方法始于正则扩扑，并识别其中可被其他线替换的线集以创建新扩扑，例如reguli、Bruen链和巢。本工作提供计算算法，以枚举在$q \le 11$的正则扩扑中所有由reguli、链和巢组成的可替换集合。附加结果包括对Baker和Ebert的(q-1)-巢的改进，该改进扩展了该无限族，以及对长期猜想（即PG(3,q)中reguli的t-巢在$t \le q$时必须可替换）的反例。
英文摘要
The theory of finite translation planes is intimately tied to the theory of spreads of PG(3,q), where a spread is a partition of PG(3,q) into $q^2+1$ pairwise disjoint lines. A traditional method of generating spreads begins with the regular spread and identifies sets of lines therein that can be replaced with other lines to create a new spread, such as reguli, Bruen chains, and nests. This work provides computational algorithms to enumerate all possible replaceable sets consisting of reguli, chains and nests in the regular spreads of PG(3,q) for $q \le 11$. Additional results include a refinement to Baker and Ebert's (q-1)-nests which expands that infinite family, and counterexamples to the long-standing conjecture that a t-nest of reguli in PG(3,q) must be replaceable if $t \le q$.
