α-Flocks and Hyperovals

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Abstract

A generalization of the concept of a flock of a quadratic cone in PG(3, q), q even, where the base of the cone is replaced by a translation oval, was introduced in [4] and is the focus of this work. The related idea of a q-clan is also generalized and studied with a particular emphasis on the connections with hyperovals. Several examples are given leading to new proofs of the existence of known hyperovals, unifying much of what has been done in this area. Finally, a proof that the Cherowitzo hyperovals do form an infinite family is also included.

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