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。

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.