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 $q=2^{ 2n+1}\geq 8$, there exists a $2$-design with parameters $(q^{ 2}+1,q,q-1)$ and the replication number $q^{ 2}$ admitting the Suzuki group $\textsf{ Sz(q)}$ 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 $\textsf{ Sz(q)}$ as their automorphism groups.
Categories: math.GR math.CO