k-radius序列及相关组合问题的研究

The existence and construction problems of combinatorial sequences satisfying certain properties are fundamental problems in the interdisciplinary research of combinatorial design theory and related fields. Motivated by some problems in computer science concerning large data transfer, k-radius sequences are a family of new combinatorial structures closely related to universal cycles and difference sets. The project focuses on the conjectures about the shortest lengths of k-radius sequences posed by Bondy et.al, and several related combinatorial problems including constructions of k-additive sequences, universal cycles and overlap cycles for block designs, and splitter sets from some tiling problems. By applying methods from combinatorial design theory, algebraic number theory and probability theory, we want to solve the conjecture posed by Bondy et.al for some fixed integers k, then solve it completely; determine the existence of universal cycles and overlap cycles for block designs, packings and coverings, and make further progress on the conjecture about universal cycles for sets posed by Chung et.al; explore the construction methods of splitter sets and related structures, and use it to solve coding problems from flash memory. The study of these problems will improve the interdisciplinary research of Combinatorics and computer science.

满足特定性质的组合序列的存在性和构造问题是组合设计理论及交叉学科研究中的一类基本问题。K-radius序列源于大数据传输技术，是和universal环及差集密切相关的一类新型组合序列。本项目围绕Bondy等人提出的关于最短k-radius序列长度的猜想，对相关组合结构，包括k-additive序列、区组设计上的universal环和overlap环，以及tiling问题中的splitter集等展开研究。希望借助组合设计理论、代数数论和概率方法等工具，证明固定整数k时Bondy等人的猜想成立，进而完全证明该猜想；确定区组设计及填充、覆盖上的universal环和overlap环的存在性，并推进Chung等人关于子集上的universal环的猜想；深入探索splitter集及相关结构的构造方法，解决闪存纠错技术中的编码问题。这些问题的研究和解决必然会丰富和加强组合学和计算机科学的交叉研究。

本项目研究了Bondy等人关于k-radius序列最短长度的两个猜想，利用Weil定理和循环填充证明了两个猜想在更多参数下是正确的；利用代数数论工具刻画了完美分裂集在某些参数下存在的充分必要条件，并给出了具体构造；利用填充设计和Ramsey定理回答了Jedwab等人关于t-suitable核的公开问题。研究计划调整后，项目组在闪存、分布式存储和DNA存储等各类数据存储的编码问题上，以及铺砖问题相关的UPB问题上做出了一系列的重要研究成果。这些成果既丰富了相关组合理论，也推进了交叉领域的发展。本项目发表基金标注论文21篇。团队成员获一项国家级青年人才奖励，担任两个国际期刊编委。培养博士毕业生3名，硕士毕业生2名，资助在站博士后1名，组织组合设计青年论坛系列活动6期。