# Transitive $t$-designs and strongly regular graphs constructed from linear groups $L(2,q)$‎, ‎$q \leq 23$

Document Type: Research Paper

Authors

Department of Mathematics, University of Rijeka, Rijeka, Croatia

Abstract

‎In this paper we construct transitive $t$-designs from the linear groups $L(2,q)‎, ‎q \leq 23$‎.
‎Thereby we classify $t$-designs‎, ‎$t \ge 2$‎, ‎admitting a transitive action of the linear groups $L(2,q)‎, ‎q \leq 23$‎, ‎up to 35 points and obtained numerous transitive designs‎, ‎for $36\leq v\leq 55$‎. ‎In many cases we proved the existence of $t$-designs with certain parameter sets‎. ‎Among others we constructed $t$-designs with parameters $2$-$(55,10,4)$‎, ‎$3$-$(24,11,495)$‎, ‎$3$-$(24,12‎, ‎5m)‎, ‎m \in \{11‎, ‎12,22‎, ‎33‎, ‎44‎, ‎66‎, ‎132\}$‎. ‎Furthermore‎, ‎we constructed strongly regular graphs admitting a transitive action of the linear groups $L(2,q)‎, ‎q \leq 23$‎.

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