可分组3-设计理论及应用研究

Combinatorial design theory is one of the important branches of Combinatorics, where we study the discrete objects with high balanced properties and nice structures according to specified rules. One of the most classical objects is t-design. Since Wilson's Fundamental Construction came out in 1960s for the construction of pairwise balanced designs, and was then extended by Hartman for the construction of 3-designs, the existence problem of group divisible 3-designs has become one of the most basic problems in combinatorial design theory, especially in the theory of 3-wise balanced designs. However, little is known about the existence of group divisible 3-designs due to the complicated structures. In this research, we will study the existence of group divisible 3-designs and related problems. Especially, we expect to pursue promising ideas of the designs of high-performance algorithms and effective direct constructions of group divisible 3-designs with small orders and all possible types, to improve the existence results of the ones with five groups and the ones with all but one group of the same size by studying the relations between designs with different types. Based on the existing constructions in the theory of 3-designs, we also study some related problems in coding theory and database theory, such as the combinatorial constructions of optimal constant composition codes and the optimal acyclic decompositions of uniform hypergraphs. These problems have recently attracted many combinatorists' attention, and the solution will contribute very well to the further development of related areas. Further, since group divisible 3-designs have close relations with fundamental combinatorial problems such as 3BD closed sets and large sets of triple systems, as well as the wide application in coding theory and computer science, our research in this proposal is of significant value from both the theoretical and applicable points of view.

组合设计是组合数学的一个重要分支，它主要研究具有高度平衡性及完美结构的离散对象,如t-设计等。当Hartman将"Wilson基本构作法"推广并用来构造3-平衡设计之后，可分组3-设计的存在性问题便成为3-设计理论研究中最基本的核心问题之一。本项目拟对可分组3-设计的存在性及相关的应用问题进行研究。理论方面主要研究型不一致的可分组3-设计的存在性，包括对小阶数所有可能的型的可分组3-设计的算法设计及直接构造，推进组个数为5的可分组3-设计的存在性结果，研究除一个组外组的大小均相等的可分组3-设计的构造方法及各类参数之间的相互联系。应用方面研究3-设计理论相关的编码及图分解问题，如最优常重复合码的组合构造方法、超图的最优无圈分解等。鉴于可分组3-设计是3-设计理论的经典设计，其研究和发展又对编码密码学和计算机科学等研究领域有积极的推动作用，从而本课题的研究具有重要的理论意义和应用价值。

本项目以组合设计理论为主要工具，结合数论、图论、有限域等数学方法，对与组合数学密切相关的几类组合编码的存在性和构造问题做了系统的研究，包括Enomoto-Katona 空间的最优纠错码、最优纠删码、多重常重码、线形大小的常重码和常重复合码等。在研究这些编码问题的过程中，对相关的组合设计做了推广，如推广的填充设计、推广的Mendelsohn设计，以及Hanani 三元填充等。同时，本项目还对完全超图的最优α-无圈分解问题、可列表解码的随机自正交码以及拟群的自同构群的性质等课题进行了研究。三年来，先后在重要国际刊物《IEEE Transactions on Information Theory》、《Combinatorics, Probability and Computing》、《Designs, Codes and Cryptography》和《The Journal of Combinatorial Designs》上发表4篇论文，另有1篇被接收。