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In this paper we construct structures from Mathieu group $M_{11}$. Weclassify transitive $t$-designs with 11, 12 and 22 points admitting atransitive action of Mathieu group $M_{11}$. Thereby we proved the existence ofdesigns with parameters 3-(22,7,18) and found first simple designs withparameters 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, weclassified strongly regular graphs on at most 450 vertices admitting atransitive action of the Mathieu group $M_{11}$.

On transitive designs and strongly regular graphs constructed from Mathieu group $M_{11}$