A note on two families of $2$-designs arose from Suzuki-Tits ovoid

Abstract

In this note, we give a precise construction of one of the families of $2$-designs arose from studying flag-transitive $2$-designs with parameters $(v,k,\lambda )$ whose replication numbers $r$ are coprime to $\lambda$. We show that for a given positive integer $q=2^{2n+1}\ge 8$, there exists a $2$-design with parameters $(q^2+1,q,q-1)$ and the replication number $q^2$ admitting the Suzuki group $\text{Sz(q)}$ as its automorphism group. We also construct a family of $2$-designs with parameters $(q^2+1,q(q-1),(q-1)(q^2-q-1))$ and the replication number $q^2(q-1)$ admitting the Suzuki groups $\text{Sz(q)}$ as their automorphism groups.