Classification of flocks of the quadratic cone in PG(3,64)

TL;DR

This paper completes the classification of flocks of the quadratic cone in PG(3,q) for q ≤ 71, identifying exactly three such flocks in PG(3,64) through computational methods and known geometric correspondences.

Contribution

It provides a complete classification of quadratic cone flocks in PG(3,64), confirming only three known types up to projective equivalence.

Findings

Exactly three flocks in PG(3,64) identified

Classification achieved via computational verification

Utilizes connection between flocks and herds of ovals

Abstract

Flocks are an important topic in the field of finite geometry, with many relations with other objects of interest. This paper is a contribution to the difficult problem of classifying flocks up to projective equivalence. We complete the classification of flocks of the quadratic cone in PG(3,q) for q <= 71, by showing by computer that there are exactly three flocks of the quadratic cone in PG(3,64), up to equivalence. The three flocks had previously been discovered, and they are the linear flock, the Subiaco flock and the Adelaide flock. The classification proceeds via the connection between flocks and herds of ovals in PG(2,q), q even, and uses the prior classification of hyperovals in PG(2,64).