A linear set view on KM-arcs

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 F 2 -linear set, called an i -club ,...

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Bibliographic Details
Published in:Journal of algebraic combinatorics 2016-08, Vol.44 (1), p.131-164
Main Authors: De Boeck, Maarten, Van de Voorde, Geertrui
Format: Article
Language:English
Subjects:
Combinatorics
Computer Science
Convex and Discrete Geometry
Group Theory and Generalizations
Lattices
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
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Summary: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 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 .
ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-015-0661-7