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.
Highlights
For larger S, it is beneficial to not explicitly verify that Ci ∩RH,Si = ∅ but instead only verify that Ci ∩Si = ∅ and only perform the full verification at the resulting leaf nodes. This way, we travel through more nodes than strictly necessary for the tree, but since only ∅ can appear as extra set in the union, this will not harm the correctness of the algorithm
For each of the known types H, we explicitly compute the H4-orbit of H, using Remark 7
It is sufficient to test for each hyperoval whether or not it belongs to one of these four H4-orbits
Summary
Conics and hyperovals are the most extensively studied objects in Desarguesian finite projective planes; they are connected to a large variety of combinatorial structures and practical applications. In 1994, Penttila and Royle [18] showed that there are exactly six classes of hyperovals in PG(2, 32): five constructions listed in Table 1 and one sporadic construction [12]. Penttila and Royle [19] classified all hyperovals in PG(2, 64) admitting a collineation of order 2 or 3. This leaves only the largest case open: the existence of a hyperoval in PG(2, 64) without any nontrivial automorphisms.
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