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

In this note, we give a precise construction ofone of the families of 2-designs arose from studying ŕag-transitive 2-designs with parameters(v, k, λ) whose replication numbersrare coprime to λ. We show that for a given positive integer q=22n+1⩾8, there exists a 2-design with parameters (q2+ 1, q, q−1) and the replication numberq 2 admitting the Suzuki group Sz(q) asits automorphism group. We also construct a family of 2-designs with parameters (q2+ 1, q(q−1),(q−1)(q2−q−1)) and thereplication number q2(q−1) admitting the Suzuki groups Sz(q) astheir automorphism groups.