循环设计以及相关编码的组合构造研究

项目名称： 循环设计以及相关编码的组合构造研究

项目编号： No.11201252

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 王小苗

作者单位： 宁波大学

项目金额： 22万元

中文摘要： 循环设计是以循环群为自同构群的一类组合设计，无论在理论上还是在通讯系统和计算机科学等领域的应用中，都有着重要作用。本项目拟探索循环设计的直接构造和递归构造方法，并将主要针对严格循环可分组设计、半循环带洞可分组设计、循环差族、完美相对差族、有向设计以及可分解设计等设计理论进行系统研究，从而扩大循环设计的构造方法和存在性结果。同时，本项目将利用组合设计的方法来构造自相关性和互相关性不等的一维和二维最优光正交码，从而建立最优光正交码存在的新的无穷类。其中，当自相关性和互相关性不等时权重为3的二维最优光正交码和权重为4的一维最优光正交码是研究的重点。此外，将给出一些特殊权重和指标的小阶数光正交码的构造结果。

中文关键词： 半循环；可分组设计；光正交码；平衡样本设计；圈分解

英文摘要： A cyclic design is a class of combinatorial designs, whose automorphism group is a cyclic group. It plays an important role not only in theory but also in the application of the communication system and computer science and other fields. The project aims to investigate the direct and recursive constructions for cyclic designs, and mainly study systematically on the design theory including strictly cyclic group divisible design, semi-cyclic holey group divisible design, cyclic difference family, perfect relative difference family, directed design and resolvable design, and subsequently extend the construction methods and existence results of cyclic designs. Meanwhile, in this project, we use combinatorial designs to construct optimal one-dimensional and two-dimensional optical orthogonal codes(OOCs) when the auto-correlation property is unequal to the cross-correlation property, and then establish new infinite classes of optimal OOCs. Particularly, when the auto-correlation property is unequal to the cross-correlation property, we focus on both optimal two-dimensional OOCs with weight three and one-dimensional OOCs with weight four. Also, we present some construction results about optical orthogonal codes of small order with special weight and index.

英文关键词： semi-cyclic；group divisible design；optical orthogonal code；balanced sampling design；cycle decomposition