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