检验q-Steiner系统的一种新方法
A New Approach to Examine q-Steiner Systems
摘要：
在$q$ -模拟设计和编码中，最有趣的问题之一是一般Steiner系统（不包括扩展）的$q$ -模拟无限族的存在性问题，特别是Fano平面的$q$ -模拟的存在性问题.我们展示了一种全新的方法来解决这个问题。在这个过程中，我们定义了一个新的家庭的设计，其存在是暗示的存在$q$ -Steiner系统，但可能存在，即使相关的$q$ -Steiner系统不存在。该方法是基于一个可能的系统，通过穿孔的所有子空间的$q$ -Steiner系统几次。我们定义的打孔系统作为一种新型的设计，并列举了各种类型的子空间，它可能有。显然，它的存在并不意味着相关的$q$ -Steiner系统的存在。另一方面，这种类型的设计展示了我们如何接近相关的$q$ -Steiner系统。给出了此类设计存在的必要条件。这些必要条件也将是相关的$q$ -Steiner系统存在的必要条件。给出了这种设计的平凡和非平凡的直接构造以及非平凡的递归构造。一些设计具有对称结构，其在系统中现有子空间的维度上是均匀的。大多数构造都是基于设计的这种统一结构或其穿孔设计。最后，基于这种新方法，考虑了任意给定$q$的$q$ -Fano平面的结构.
英文摘要：
One of the most intriguing problems, in $q$-analogs of designs and codes, is the existence question of an infinite family of $q$-analog of Steiner systems (spreads not included) in general, and the existence question for the $q$-analog of the Fano plane in particular. We exhibit a completely new method to attack this problem. In the process we define a new family of designs whose existence is implied by the existence of $q$-Steiner systems, but could exist even if the related $q$-Steiner systems do not exist. The method is based on a possible system obtained by puncturing all the subspaces of the $q$-Steiner system several times. We define the punctured system as a new type of design and enumerate the number of subspaces of various types that it might have. It will be evident that its existence does not imply the existence of the related $q$-Steiner system. On the other hand, this type of design demonstrates how close can we get to the related $q$-Steiner system. Necessary conditions for the existence of such designs are presented. These necessary conditions will be also necessary conditions for the existence of the related $q$-Steiner system. Trivial and nontrivial direct constructions and a nontrivial recursive construction for such designs, are given. Some of the designs have a symmetric structure, which is uniform in the dimensions of the existing subspaces in the system. Most constructions are based on this uniform structure of the design or its punctured designs. Finally, the structure of the $q$-Fano plane for any given $q$, was considered based on this new approach.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
