Combinatorial designs and their generalizations

本研究旨在提高我们对组合设计及其推广的理解。在最基本的情况下，基于某个基集的设计是一个子集的集合，这些子集以相等的频率覆盖任何两个不同的元素。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一条线。设计是著名谜题的基础，例如数独和“囚犯的帽子问题”。更重要的是，设计本质上在信息论（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）中是有用的。这些应用程序把重点放在有限设计上，而由有限域（例如二进制序列）产生的美丽几何形成了一个自然的起点。

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.