A $\left(k,l\right)$‐configuration is a set of $l$ blocks on $k$ points. For Steiner triple systems, $\left(l+2,l\right)$‐configurations are of particular interest. The smallest nontrivial such configuration is the Pasch configuration, which is a $\left(6,4\right)$‐configuration. A Steiner triple system of order $v$, an STS $\left(v\right)$, is $r$‐sparse if it does not contain any $\left(l+2,l\right)$‐configuration for $4\le l\le r$. The existence problem for anti‐Pasch Steiner triple systems has been solved, but these have been classified only up to order 19. In the current work, a computer‐aided classification shows that there are 83,003,869 isomorphism classes of anti‐Pasch STS(21)s. Exploration of the classified systems reveals that there are three 5‐sparse STS(21)s but no 6‐sparse STS(21)s. The anti‐Pasch STS(21)s lead to 14 Kirkman triple systems, none of which is doubly resolvable.

中文翻译：

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稀有Steiner 21阶三元系统

一种 $（ķ，升）$-配置是一组 $升$ 阻止 $ķ$点。对于Steiner三重系统，$（升+2，升）$－配置特别重要。最小的此类平凡配置是Pasch配置，即$（6，4）$-组态。斯坦纳三阶系统$v$，一个STS $（v）$，是 $\mathrm{[R}$-如果不包含任何内容，则稀疏 $（升+2，升）$-配置 $4\le 升\le \mathrm{[R}$。反Pasch Steiner三元系统的存在问题已得到解决，但仅分类到19级。在目前的工作中，计算机辅助分类显示，反Pasch STS有83,003,869个同构类（21）。 s。对分类系统的探索表明，存在三个5稀疏STS（21），而没有6稀疏STS（21）。反Pasch STS（21）导致14个柯克曼三元系统，没有一个是双重可分辨的。