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Algebraic Constructions of Complete m-Arcs Daniele Bartoli, Giacomo Micheli
Combinatorica ( IF 1.1 ) Pub Date : 2022-01-14 , DOI: 10.1007/s00493-021-4712-5
Daniele Bartoli 1 , Giacomo Micheli 2
Affiliation  

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.



中文翻译:

完整 m 弧的代数结构 Daniele Bartoli, Giacomo Micheli

m为正整数,q为素数幂,PG(2, q ) 为有限域\({\mathbb{F}_q}\)上的射影平面。在大小小于q的 PG(2, q ) 中找到完整的m弧是有限几何中的经典问题。在本文中,当q与m相比相对较大时,我们给出了这个问题的完整答案,明确地构造了迄今为止文献中对于任何m ≥ 8的最小m弧。对于任何固定的m,我们的弧\({{\ cal A}_{q,m}}\)满足\(\left| {{{\cal A}_{q,m}}} \right| - q \to - \infty \)随着q的增长。为了产生这样的m-弧,我们开发了一种伽罗瓦理论机制,它允许将弧外点的几何信息传递给算术信息,这反过来又可以证明弧的m-完备性。

更新日期:2022-01-16
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