A S(3, {4, 6}, 18) with a subdesign S(3, 4, 8) does not exist

For $K \subset \{1,2,3,\ldots\}$ , a S ( t , K , v ) design is a pair, $(V,{\cal B})$ , with | V | =  v and ${\cal B}$ a set of subsets of V such that each t ‐subset of V is contained in a unique $\alpha \in {\cal B}$ and $\vert \alpha \vert \in K$ for all $\alpha \in {\cal B}$ . If $U \subseteq V$ , $\vert U\vert=u$ , ${\cal A}=\{\alpha\in{\cal B} : \alpha\subseteq U\}$ , and $(U,{\cal A})$ is a S ( t , K , u ) design, then we say $(V,{\cal B})$ has a subdesign on U . We show that a S (3,{4,6},18) design with a subdesign S (3,4,8) does not exist. © 2007 Wiley Periodicals, Inc. J Combin Designs 17: 36–38, 2009