具有最多不动点的金字塔型阿达马设计的刻画
Characterizing pyramidal Hadamard designs with the largest number of fixed points
浏览论文内容
中文总结 AI 辅助
本文刻画具有最多不动点的金字塔型阿达马设计，推广了此前仅针对特定阿贝尔群及$m$为2的幂的结果，证明相关设计族与存在指定金字塔型自同构群的阿达马设计族一致。
中文摘要 AI 辅助
对称$(v,k,\lambda)$-设计在群$G$作用下被称为$f$-金字塔型（其中$f<v-1$），当且仅当$G$作为自同构群固定$f$个点，并在其余点上严格传递作用。本文证明必然有$f\leq v -2(k-\lambda)$；特别地，对于参数为$(v,k,\lambda)=(4m-1, 2m-1, m-1)$（$m$为自然数）的阿达马设计，可得$f\leq 2m-1$。近期，针对补设计的研究已完全确定：当$m=2^{k}>1$时，存在阿贝尔$(2m-1)$-金字塔型自同构群$G$的阿达马$(4m-1,2m-1,m-1)$设计族$P$。本文将该结果推广，证明$P$与所有存在$(2m-1)$-金字塔型自同构群的阿达马$(4m-1,2m-1,m-1)$设计族一致，且无需假设群为阿贝尔或$m$为2的幂。
英文摘要
A symmetric $(v,k,λ)$-design is said to be $f$-pyramidal, with $f<v-1$, under the action of a group $G$ if $G$ acts as an automorphism group fixing $f$ points and acting sharply transitively on the remaining ones. We show that, necessarily, $f\leq v -2(k-λ)$. In particular, for a Hadamard design with parameters $(v,k,λ)=(4m-1, 2m-1, m-1)$, for some $m\in\mathbb{N}$, it follows that $f\leq 2m-1$. Recently, working on the complement design, the family $P$ of Hadamard $(4m-1,2m-1,m-1)$-designs admitting an abelian $(2m-1)$-pyramidal automorphism group $G$ has been completely determined in the case $m=2^{k}>1$. In this paper, we generalize that result by showing that $P$ coincides with the family of Hadamard $(4m-1,2m-1,m-1)$-designs admitting a $(2m-1)$-pyramidal automorphism group, without assuming either that the group is abelian or that $m$ is a power of $2$.
