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

## TL;DR

This paper constructs and classifies combinatorial designs and strongly regular graphs derived from the Mathieu group M11, revealing new existence results and classifications for these mathematical structures.

## Contribution

It introduces new transitive designs with specific parameters and classifies strongly regular graphs with M11 symmetry up to 450 vertices.

## Key findings

- Existence of 3-(22,7,18) designs.
- First simple 4-(11,5,6) and 5-(12,6,6) designs.
- Classification of strongly regular graphs with M11 symmetry up to 450 vertices.

## Abstract

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 $M_{11}$.