Ree groups as automorphism groups of block designs

A recent classification of flag-transitive 2-designs with parameters $(v,k,\lambda )$ whose replication number $r$ is coprime to $\lambda$ gives rise to eight possible infinite families of 2-designs, some of which are with new parameters. In this note, we give explicit constructions for two of these families of 2-designs, and show that for a given positive integer $q=3^{2n+1}⩾27$, there exist 2-designs with parameters $(q^3+1,q^i,q^i-1)$, for $i=1,2$, admitting the Ree group ${}^2{G}_2\left(q\right)$ as their automorphism groups.