Combinatorial designs and their generalizations

本研究的目的是提高我们对组合设计和它们的一些概括的理解。在最基本的情况下，某个基础集上的设计是子集的集合，这些子集经常覆盖任意两个不同的元素。这在许多几何模型中是一个熟悉的概念，因为两个不同的点唯一地确定了一条线。设计是众所周知的谜题的基础，比如数独和“囚犯的帽子问题”。更重要的是，就其本质而言，设计在信息论（通过与纠错码的联系）、计算机科学（软件测试、网络设计）和统计学（实验设计）中都是有用的。这些应用程序将重点放在有限的设计上，而由有限域（例如二进制序列）产生的美丽几何形状形成了一个自然的起点。******可以用多种方式扩展上面的基本定义。在t型设计中，任意t个不同的点将包含在相同数量的块中。在图分解中，对之间的边或连接将被划分为给定小结构的副本。进一步可能的扩展包括使用边缘颜色来模拟两个或多个同时发生的关系。举个玩具的例子，一场桥牌锦标赛可能要求每一对玩家恰好成为搭档一次，恰好成为对手两次。******虽然有许多已知的特殊结构，但设计和图形分解的一般存在性问题是出了名的困难。本研究主要关注具有挑战性和最普遍的情况，特别是在标准方法不适用的情况下。******最近一个关于设计存在性的著名定理把这个话题带到了组合数学的前沿。即使这个理论对于非常大的设计是完整的，仍然有相当多的工作要做。特别是，特定设计不存在的原因导致了与数学其他领域的奇妙联系，包括代数、分析和几何。******与应用数学和其他科学相比，这项研究离直接的工业应用还有一段距离。然而，与此相抵消的是，它足够通用，可以在各种各样的应用程序中具有最终用途。此外，基于信息的问题对设计、代码或数组施加的额外结构通常在数学上是很自然的。通过这种方式，研究紧密地以其应用为指导。

This research aims to improve our understanding of combinatorial designs and some of their generalizations. In the most basic case, a design on some ground set is a collection of subsets which covers any two distinct elements equally often. This is a familiar idea in many geometric models, since two different points uniquely determine a line. Designs underlie well-known puzzles, such as Sudoku and the “prisoner's hat problem”. More importantly, by their nature designs are useful in information theory (through their connection with error-correcting codes), computer science (software testing, network design) and statistics (experimental design). These applications have placed the focus on finite designs, and the beautiful geometries arising from finite fields (binary sequences, for instance) form a natural starting point.******It is possible to extend the basic definition above in a number of ways. In a t-design, any t distinct points are to be contained in the same number of blocks. In a graph decomposition, the edges or connections between pairs are to be partitioned into copies of a given small structure. A further possible extension includes the use of edge colours to model two or more simultaneous relationships. As one toy example, a bridge tournament might require every pair of players to be partners exactly once and opponents exactly twice.******Although there are many special constructions known, the general existence question for designs and graph decompositions is notoriously hard. This research is mainly concerned with the challenging and most general cases, and especially in situations where standard methods don't apply.******A recent celebrated theorem on existence of designs has brought this topic to the forefront of combinatorial mathematics. Even as this theory is complete for extremely large designs, there is still considerable work to be done. In particular, the reasons why specific designs fail to exist leads to wonderful connections to other areas of mathematics, including algebra, analysis and geometry.******In comparison with applied mathematics and other sciences, this research is admittedly a step removed from direct industrial applications. However, offsetting this, it is sufficiently general to have end-use in a wide variety of applications. Moreover, the additional structure that information-based problems impose on designs, codes, or arrays is quite often mathematically natural. In this way, the research is guided closely by its applications.