An elementary description of the Mathieu dual hyperoval and its splitness

Abstract

An elementary new construction of a $3$-dimensional dual hyperoval $\mathcal{ℳ}$ over ${\mathbb{F}}_4$ is given, as well as an explicit analysis of the structure of its automorphism group. This provides a self-contained introduction to the Mathieu simple group $M_{22}$. The basic properties of $\mathcal{ℳ}$ as a dimensional dual hyperoval, e.g. splitness, complements, linear systems, quotients and coverings, are derived from this construction.