Abstract
Let m be a positive integer, q be a prime power, and PG(2,q) be the projective plane over the finite field \({\mathbb{F}_q}\). Finding complete m-arcs in PG(2,q) of size less than q is a classical problem in finite geometry. In this paper we give a complete answer to this problem when q is relatively large compared with m, explicitly constructing the smallest m-arcs in the literature so far for any m ≥ 8. For any fixed m, our arcs \({{\cal A}_{q,m}}\) satisfy \(\left| {{{\cal A}_{q,m}}} \right| - q \to - \infty \) as q grows. To produce such m-arcs, we develop a Galois theoretical machinery that allows the transfer of geometric information of points external to the arc, to arithmetic one, which in turn allows to prove the m-completeness of the arc.
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Acknowledgments
The research of D. Bartoli was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA — INdAM). In addition, this material is based upon work supported by the National Science Foundation under Grant No. 2127742 (PI: G. Micheli). Part of this work was done while the first author was visiting the University of South Florida. We thank the referees for their valuable comments and suggestions to improve the paper’s quality.
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Bartoli, D., Micheli, G. Algebraic Constructions of Complete m-Arcs. Combinatorica 42, 673–700 (2022). https://doi.org/10.1007/s00493-021-4712-5
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DOI: https://doi.org/10.1007/s00493-021-4712-5