Elation KM-Arcs

Abstract

In this paper, we study KM-arcs in PG(2, q), the Desarguesian projective plane of order q. A KM-arc \(\mathcal{A}\) of type t is a natural generalisation of a hyperoval: it is a set of q+t points in PG(2, q) such that every line of PG(2, q) meets \(\mathcal{A}\) in 0, 2 or t points.

We study a particular class of KM-arcs, namely, elation KM-arcs. These KM-arcs are highly symmetrical and moreover, many of the known examples are elation KM-arcs. We provide an algebraic framework and show that all elation KM-arcs of type q/4 in PG(2, q) are translation KM-arcs. Using a result of [2], this concludes the classification problem for elation KM-arcs of type q=4.

Furthermore, we construct for all q = 2h, h > 3, an infinite family of elation KM-arcs of type q/8, and for q=2h, where 4, 6, 7 | h an infinite family of KM-arcs of type q/16. Both families contain new examples of KM-arcs.

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Correspondence to Geertrui van de Voorde.

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This author is supported by the BOF (Special Research Fund) of Ghent University.

This author is a postdoctoral fellow of the Research Foundation — Flanders (FWO).

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de Boeck, M., van de Voorde, G. Elation KM-Arcs. Combinatorica 39, 501–544 (2019). https://doi.org/10.1007/s00493-018-3806-1

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Mathematics Subject Classification (2010)

  • 51E20
  • 51E21