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

TL;DR

This paper constructs specific 2-designs with Suzuki group automorphisms, detailing their parameters and showing their existence for certain powers of 2, advancing the understanding of symmetric combinatorial designs.

Contribution

It provides explicit constructions of two families of 2-designs with Suzuki group automorphisms, expanding the known classes of such symmetric designs.

Findings

Existence of 2-designs with parameters (q^2+1, q, q-1) for q=2^{2n+1}≥8

Construction of 2-designs with parameters (q^2+1, q(q-1), (q-1)(q^2 - q - 1))

Automorphism groups of these designs are the Suzuki groups Sz(q)

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 $\textsf{Sz(q)}$ 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.