Abstract
In this paper, we study KM-arcs of type t, i.e., point sets of size \(q+t\) in \(\mathrm {PG}(2,q)\) 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 \(\mathbb {F}_2\)-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 \(q/4\).
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Maarten De Boeck: This author is supported by the BOF-UGent (Special Research Fund of Ghent University).
Geertrui Van de Voorde: This author is a postdoctoral fellow of the Research Foundation Flanders (FWO—Vlaanderen).
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De Boeck, M., Van de Voorde, G. A linear set view on KM-arcs. J Algebr Comb 44, 131–164 (2016). https://doi.org/10.1007/s10801-015-0661-7
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DOI: https://doi.org/10.1007/s10801-015-0661-7