A linear set view on KM-arcs

Abstract

In this paper, we study KM-arcs of type t, i.e., point sets of size q+ t in PG (2 , q) such that every line contains 0, 2 or t of its points. We use field reduction to give a different point of view on the class of translation arcs. Starting from a particular F2-linear set, called an i-club, we reconstruct the projective triads, the translation hyperovals as well as the translation arcs constructed by Korchmáros-Mazzocca, Gács-Weiner and Limbupasiriporn. We show the KM-arcs of type q/ 4 recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a family of KM-arcs of type q/ 4. We show that this family, apart from new examples that are not translation KM-arcs, contains all translation KM-arcs of type q/ 4.

Publication Title

Journal of Algebraic Combinatorics

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