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 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 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.
Publication Title
Combinatorica
Recommended Citation
de Boeck, M., & van de Voorde, G. (2019). Elation KM-Arcs. Combinatorica, 39 (3), 501-544. https://doi.org/10.1007/s00493-018-3806-1
