Classification of the Hyperovals in PG(2,64)

Abstract

In this paper, we present a full classification of the hyperovals in the finite projective plane $\mathrm{PG}(2,64)$, showing that there are exactly 4 isomorphism classes. The techniques developed to obtain this result can be applied more generally to classify point sets with $0$ or $2$ points on every line, in a broad range of highly symmetric incidence structures.

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

“…Our aim in this section is to classify all flocks of the quadratic cone in PG (3,64), by bringing to bear our knowledge of ovals of PG(2, 64), based on Vandendriessche's determination of the hyperovals of PG(2, 64) [57]. It is clear that the most convenient model to approach this classification from this perspective is that of herds of ovals of PG (2,64).…”

“…Proof. In [57] it was shown that, up to equivalence, there are exactly four hyperovals in PG(2, 64): the regular hyperoval, the Adelaide hyperoval, the Subiaco I hyperoval, the Subiaco II hyperoval. The corresponding o-polynomials can be read off from the q-clans above: t 1/2 gives the regular hyperoval, f 0 from the Adelaide q-clan C A gives the Adelaide hyperoval, f 0 from the Subiaco q-clan C S gives the Subiaco I hyperoval and f a from herd H(C S ) arising from the Subiaco q-clan gives the Subiaco II hyperoval, where a 2 + a + 1 = 0.…”

“…By the correspondence between flocks of the quadratic cone in PG(3, q) and herds of ovals in PG(2, q) for q even, this is equivalent to classifying the flocks of the quadratic cone in PG (3,64), up to equivalence. That we are able to do this depends upon Vandendriessche's recent determination of the hyperovals of PG(2, 64) [57]. This is a second reason to approach this problem via herds: applying the knowledge of hyperovals is easiest in the herd setting.…”

Uniqueness of the inversive plane of order sixty-four

“…The hyperoval with a group of order 12 was generalised to the infinite family of Adelaide hyperovals in [9] in 2003. In 2019 [40], hyperovals of PG(2, 64) were classified by Vandendriessche.…”

Restricted Variable Chevalley-Warning Theorems

“…In particular, certain geometrical objects in finite projective planes known as hyperovals (see [BCP06] and the references therein for the list of known infinite families of hyperovals), give rise to such Vandermonde sets. There are further examples of Vandermonde sets known for small fields (see for example [AH19, Example 7]), but a full classification is out of reach (simply because a full classification of hyperovals appears to be out of reach [Va19]). We ask the following question related to the enumeration of Vandermonde sets.…”