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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 showthat for a given positive integer $q=2^{2n+1}\geq 8$, there exists a $2$-designwith parameters $(q^{2}+1,q,q-1)$ and the replication number $q^{2}$ admittingthe Suzuki group $\textsf{Sz(q)}$ as its automorphism group. We also constructa family of $2$-designs with parameters $(q^{2}+1,q(q-1),(q-1)(q^{2}-q-1))$ andthe replication number $q^{2}(q-1)$ admitting the Suzuki groups$\textsf{Sz(q)}$ as their automorphism groups.

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