$4-(21,5,\lambda )$ Designs from a Group of Order $171$

Abstract

Using basis reduction, we settle the existence problem for $4$-$(21,5,\lambda )$ designs with $\lambda \in \{3,5,6,8\}$. These designs each have as an automorphism group the Frobenius group $G$ of order $171$ fixing two points. We also show that a $4$-$(21,5,1)$ design cannot have the subgroup of order $57$ of $G$ as an automorphism group.