论文标题
完整$ M $ -Arcs的代数结构
Algebraic constructions of complete $m$-arcs
论文作者
论文摘要
令$ m $为正整数，$ q $是主要功率，而$ \ mathrm {pg}（2，q）$是有限字段$ \ mathbb f_q $上的投射平面。在$ \ mathrm {pg}（2，q）$小于$ q $的$ \ mathrm {pg}（2，q）中查找完整的$ m $ - arcs是有限几何的经典问题。在本文中，当$ Q $与$ M $相比相对较大时，我们给出了一个完整的答案，该$ M $相对较大，迄今为止，对于任何$ M \ geq 8 $，明确构建了文献中最小的$ m $ -4 $ -Arcs。对于任何固定的$ m $，我们的arcs $ \ mathcal a_ {q，m} $满足$ | \ mathcal a_ {q，m} | -q \ rightArrow -\ rightarrow -\ infty $ as $ q $ a $ $ $ $ agros。为了产生这样的$ m $ - arcs，我们开发了一个GALOIS理论机制，该机械允许将弧外部点的几何信息传输到算术上，从而允许证明弧线的$ m $ $ $ completentes。
Let $m$ be a positive integer, $q$ be a prime power, and $\mathrm{PG}(2,q)$ be the projective plane over the finite field $\mathbb F_q$. Finding complete $m$-arcs in $\mathrm{PG}(2,q)$ 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 $m\geq 8$. For any fixed $m$, our arcs $\mathcal A_{q,m}$ satisfy $|\mathcal A_{q,m}|-q\rightarrow -\infty$ 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.
