Enumeration of All Simple $t–(t+7,t+1,2)$ Designs

Abstract

We enumerate by computer algorithms all simple $t-(t+7,t+1,2)$ designs for $1\le t\le 5$, i.e., for all possible $t$. This enumeration is new for $t\ge 3$. The number of nonisomorphic designs is equal to $3,13,27,1$ and $1$ for $t=1,2,3,4$ and $5$, respectively. We also present some properties of these designs, including orders of their full automorphism groups and resolvability.