论文标题
最小尺寸线性集的分类和结构
Classifications and constructions of minimum size linear sets
论文作者
论文摘要
本文旨在研究投影线中最小尺寸的线性集，即$ \ mathbb {f} _q $ - $ k $ in $ \ mathrm {pg}（1，q^n）的等级$ k $的线性套件承认一个重量点，并且具有一个size $ q^{k-1}+1 $。 Lunardon和第二作者（2000）以及Jena和Van de Voorde（2021）发现了这些线性集的示例。但是，最小尺寸线性集的分类结果仅以$ k \ leq 5 $而闻名。在本文中，我们为那些$ l_u $的分类结果提供了互补权重的两个点。我们构建了新的示例，还研究了相关的$ \ mathrm {γl}（2，q^n）$ - 等价问题。这些结果解决了Jena和Van de Voorde提出的一个空旷的问题。主要工具依靠Bachoc，Serra和Zémor（2017和2018）的两个结果，对Kneser和Vosper定理的线性类似物进行了类似。然后，我们总结了指出关键对和线性集之间的联系的纸张，还为关键对获得了一些分类结果。
This paper aims to study linear sets of minimum size in the projective line, that is $\mathbb{F}_q$-linear sets of rank $k$ in $\mathrm{PG}(1,q^n)$ admitting one point of weight one and having size $q^{k-1}+1$. Examples of these linear sets have been found by Lunardon and the second author (2000) and, more recently, by Jena and Van de Voorde (2021). However, classification results for minimum size linear sets are known only for $k\leq 5$. In this paper we provide classification results for those $L_U$ admitting two points with complementary weights. We construct new examples and also study the related $\mathrm{ΓL}(2,q^n)$-equivalence issue. These results solve an open problem posed by Jena and Van de Voorde. The main tool relies on two results by Bachoc, Serra and Zémor (2017 and 2018) on the linear analogues of Kneser's and Vosper's theorems. We then conclude the paper pointing out a connection between critical pairs and linear sets, obtaining also some classification results for critical pairs.
