A linear set view on KM-arcs

TL;DR

This paper offers a new perspective on KM-arcs of type t in projective geometry using field reduction, reconstructs known arcs, and introduces a family of KM-arcs of type q/4, including new examples.

Contribution

It provides a novel field reduction approach to study KM-arcs, reconstructs known arcs from linear sets, and introduces a new family of KM-arcs of type q/4.

Findings

KM-arcs of type q/4 are translation arcs

A new family of KM-arcs of type q/4 is constructed

All translation KM-arcs of type q/4 are contained in this family

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 Korchmaros-Mazzocca, Gacs-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.