Clubs of rank k are well-celebrated objects in finite geometries introduced by Fancsali and Sziklai in 2006. After the connection with a special type of arcs known as KM-arcs, they renewed their interest. This paper aims to study clubs of rank n in PG$(1,q^n)$. We provide a classification result for (n-2)-clubs of rank n, we analyze the $\mathrm{\Gamma L}(2,q^n)$-equivalence of the known subspaces defining clubs, for some of them the problem is then translated in determining whether or not certain scattered spaces are equivalent. Then we find a polynomial description of the known families of clubs via some linearized polynomials. Then we apply our results to the theory of blocking sets, KM-arcs, polynomials and rank metric codes, obtaining new constructions and classification results.
翻译：k 级俱乐部是Fancsali 和 Sziklai 于2006年推出的有限地貌中精心标注的物体。 在连接了被称为 KM-arcs 的特殊类型的弧体之后, 它们又恢复了兴趣。 本文的目的是研究n 级俱乐部, 单位为 PG$(1, q ⁇ n) 。 我们为n 级俱乐部提供了分类结果( n-2)- clubs of n级, 我们分析了 $\ mathrm\ Gamma L} ( 2, q ⁇ n) $- 等值的已知的子空间定义俱乐部, 其中一些在确定某些分散的空间是否相等时就翻译了问题。 然后我们通过一些线性多面性多面性多面体对俱乐部已知的家族进行多面描述。 然后我们将我们的结果应用到阻塞装置、 KM-arcs、 聚点和等级标准代码的理论, 获得新的构造和分类结果 。
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