Simple $3$-designs of $\text{PSL}(2,q)$ with Block Size $7$

Abstract

This paper devotes to the investigation of $3$-designs admitting the special projective linear group $\text{PSL}(2,q)$ as an automorphism group. When $q\equiv 3(\mathrm{mod}4)$, we determine all the possible values of $\lambda$ in the simple $3$-$(q+1,7,\lambda )$ designs admitting $\text{PSL}(2,q)$ as an automorphism group.