Transitive $t$-designs and strongly regular graphs constructed from linear groups $L(۲,q)$‎, ‎$q \leq ۲۳$

‎In this paper we construct transitive $t$-designs from the linear groups $L(۲,q)‎, ‎q \leq ۲۳$‎. ‎Thereby we classify $t$-designs‎, ‎$t \ge ۲$‎, ‎admitting a transitive action of the linear groups $L(۲,q)‎, ‎q \leq ۲۳$‎, ‎up to ۳۵ points and obtained numerous transitive designs‎, ‎for $۳۶\leq v\leq ۵۵$‎. ‎In many cases we proved the existence of $t$-designs with certain parameter sets‎. ‎Among others we constructed $t$-designs with parameters $۲$-$(۵۵,۱۰,۴)$‎, ‎$۳$-$(۲۴,۱۱,۴۹۵)$‎, ‎$۳$-$(۲۴,۱۲‎, ‎۵m)‎, ‎m \in \{۱۱‎, ‎۱۲,۲۲‎, ‎۳۳‎, ‎۴۴‎, ‎۶۶‎, ‎۱۳۲\}$‎. ‎Furthermore‎, ‎we constructed strongly regular graphs admitting a transitive action of the linear groups $L(۲,q)‎, ‎q \leq ۲۳$‎.

Keywords:

$t$-design‎ , ‎strongly regular graph‎ , ‎linear group‎ , ‎Transitive group