  In this paper we construct structures from Mathieu group $M_{11}$. We
classify transitive $t$-designs with 11, 12 and 22 points admitting a
transitive action of Mathieu group $M_{11}$. Thereby we proved the existence of
designs with parameters 3-(22,7,18) and found first simple designs with
parameters 4-(11,5,6) and 5-(12,6,6). Additionally, we proved the existence of
$2$-designs with certain parameters having 55 and 66 points. Furthermore, we
classified strongly regular graphs on at most 450 vertices admitting a
transitive action of the Mathieu group $M_{11}$.
