有限极空间相关的几何结构研究

In this project we propose to study the following two research topics. First, the existence and constructions of geometric structures in finite polar spaces, including the nonexistence in high rank cases and construction and classification in low rank cases. We will focus on intriguing sets and m-systems, which are natural generalizations of ovoids, spreads and hemisystems. Those are extensively studied geometric objects, and can be used to construct geometric structures like partial geometry, strongly regular graphs and association schemes with special properties. Second, the constructions and classifications of rank two incidence geometries with emphasis on projective planes and generalized quadrangles. The former is based on the work of Dembowski and Piper on the classification of finite projective planes with large quasiregular automorphism groups, and the latter is based on the coset geometries and BLT sets in Q(4,q). The projective and polar spaces of rank at least three must be classical, and the classification in the rank two case is beyond reach. It is of great theoretical interest to construct new examples and give classification under proper conditions. We will view these objects from a new perspective to make progress of our own feature. We will combine geometric insights and tools from algebra and number theory and try to make progress on some important problems. The objects we study in this project are all finite.

本项目主要研究以下两方面的内容：（1）有限极空间中以intriguing set和m-system为代表的几何结构的存在性和构造，包括高秩情形的非存在性和低秩情形的构造和分类。它们是卵形体、spread和半线系等概念的自然推广，是有限几何中得到广泛研究的几何对象，可用于构造部分几何等关联几何结构、强正则图和具有特殊性质的结合方案。（2）以射影平面和广义四边形为代表的、秩为2的关联几何的构造和分类。前者是基于Dembowski和Piper对具有较大拟正则自同构群的射影平面的分类，后者是基于陪集几何和Q(4,q)中的BLT集。秩至少是3的射影空间和极空间一定是经典的，但秩为2的情形的分类遥不可及，构造新的例子和在适当条件下的分类具有重要的理论价值。本项目将充分结合几何直观、代数和数论工具，通过新的视角审视这些问题，做出具有特色的工作，争取在若干重要问题上取得进展。本项目研究的对象都是有限的。

本项目共发表SCI论文11篇，其中包括一篇Advances in Mathmatics、一篇Combinatorica、两篇Journal of Combinatorial Theory, Series A，另有两篇发表在国际信息论权威期刊IEEE Transactions on Information Theory。本项目综合利用有限域算术性质、复杂的指数和运算和典型群子群结构，在有限极空间中几何构型的构造方面发展了一套系统的方法，利用该方法成功构造了Cameron-Liebler线族偶特征下的第一族无穷类，并在高秩辛空间中构造出m-卵形体。本项目完成了有限厄米特空间中的传递卵形体的分类工作；完全确定了奇特征下经典广义四边形W(q)的导出四边形的正则子群结构，证明其可以具有任意大的幂零类。本项目将组合方法用于代数编码研究，利用组合方法推广了Donoho-Stark不确定性原则，用部分差集给出新的极小线性码的构造。