论文标题
高度对称的柯克曼三重系统的第一批家庭，其命令填补了一致性课程
The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class
论文作者
论文摘要
Kirkman Triple Systems（KTSS）是最受欢迎的组合设计之一，它们的存在已经很久以前解决了。然而，与Steiner三重系统相比，对其自动形态群体知之甚少。特别是，没有已知的一致性类别代表具有许多自动形态的KT的顺序，至少至少接近点数。我们通过证明每当$ v \ equiv 39 $（mod 72）或$ v \ equiv 4^e48 + 3 $（mod $ 4^e96 $）和$ e \ geq 0 $时，我们就会填补这一空白。 这只是对KTSS进行仔细调查的后果之一，其自动形态组$ g $在除三分之外的所有方面都迅速运转。我们的方法都是建设性的，并且产生了KTSS，在许多情况下，它们继承了$ g $的一些自动形态，从而增加了对称的总数。 为了获得这些结果，有必要引入新型的差异家庭（双方不相交的差异）和差异矩阵（可分布的矩阵），我们认为这很有趣。
Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We fill this gap by proving that whenever $v \equiv 39$ (mod 72), or $v \equiv 4^e48 + 3$ (mod $4^e96$) and $e \geq 0$, there exists a KTS on $v$ points having at least $v-3$ automorphisms. This is only one of the consequences of a careful investigation on the KTSs with an automorphism group $G$ acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of $G$, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.
