Extensions of $\mathrm{GQ}(4,2)$, the description of hyperovals

A subgraph of the point graph of the generalized quadrangle $\mathrm{GQ}(s,t)$ is called a hyperoval, if it is a regular graph without triangles of valence $t+1$ with even number of vertices. In the triangular extensions of $\mathrm{GQ}(s,t)$ the role of $\mu$-subgraphs can be played by hyperovals only. We give a classification of the hyperovals in $\mathrm{GQ}(4,2)$. For any even $\mu$ from 6 to 18 there exists a hyperoval with $\mu$ points.
This research was supported by the Russian Foundation for Basic Research, grant 94–01–00802–a.