Hemisystems of small flock generalized quadrangles

TL;DR

This paper provides a comprehensive computer classification of hemisystems in small flock generalized quadrangles, introduces potential new infinite families, and expands understanding of their structure in known cases.

Contribution

It offers the first complete classification for certain small cases and suggests new infinite families of hemisystems in classical generalized quadrangles.

Findings

Complete classification of hemisystems in order (25,5) quadrangles.

Numerous examples of hemisystems in quadrangles up to order (121,11).

Identification of two potential new infinite families in classical quadrangles.

Abstract

In this paper, we describe a complete computer classification of the hemisystems in the two known flock generalized quadrangles of order $(5^2,5)$ and give numerous further examples of hemisystems in all the known flock generalized quadrangles of order $(s^2,s)$ for $s \le 11$. By analysing the computational data, we identify two possible new infinite families of hemisystems in the classical generalized quadrangle $H(3,q^2)$.