两类码的最优性和组合编制研究

In the past few decades, the rapid development of mathematics has further established the basic and leading status in the whole area of science and technology. As the main branch of discrete mathematics, combinatorial mathematics has played a great role in the technical innovation of modern communication field. By means of combinatorial configurations, the present project will focus on two classes of hot codes in modern communication area: equitable symbol weight codes (ESWCs) and multilength optical orthogonal codes (MLOOCs). ESWCs are a class of new codes proposed by Chee et al. (2013), which have nice performance agaist permanent narrowband noise in power line communications. Research on MLOOCs was motivated by an application in optical CDMA multimedia networks, which support multimedia service with different signaling rates and quality of service requirements. Quite recently, the present applicant with his collaborators revealed an equivalence between ESWCs and generalized balanced tournament packings and an equivalence between MLOOCs and a compatible cyclic difference packing set. This project intends to make an in-depth and systematic study on the combinatorial properties, structure characters, construction methods and optimality of aforementioned combinatorial configurations. The goal of this project is to acquire siganificant research progress on the optimality, combinatorial encoding and related problems regarding ESWCs and MLOOCs.

过去几十年中，数学科学的快速发展进一步确立了它在整个科学技术领域中的基础和主导地位。而组合数学作为离散数学的一个重要分支，对现代通信领域的技术革新发挥了巨大的作用。本项目拟通过组合构型来研究现代通信领域中的两个热点码类：符号等重码（ESWCs）和多长度光正交码（MLOOCs）。ESWCs是由Chee等（2013）引入的一类新的码，其在电力线通信中具有很强的抗窄带噪声的能力；而关于 MLOOCs 的研究源于码分多址光纤网络（OCDMA）的应用，其可以同时满足不同信号速率和多种媒体的传输需求。最近，申请人与他的合作者揭示了ESWCs与广义平衡竞赛填充之间的对等关系以及MLOOCs与相容的循环差填充簇之间的对等关系。本项目拟对上述两类组合构型的组合性质、结构特征、构造方法和最优性展开系统、深入的探讨。项目的目标是在ESWCs和MLOOCs的最优性和组合编制及其相关问题上取得重要的进展。

随着信息科学和计算机技术的快速发展，数学和信息科学之间的深刻联系正在发挥着越来越重要的作用。本项目瞄准信息科学前沿，重点研究现代通信领域中与组合数学密切相关的两个热点码类，即码分多址光纤网络中的多长度光正交码（MLOOC）和电力线通信中的符号等重码（ESWC）。本项目借助于MLOOCs和ESWCs的组合特性，对MLOOCs和ESWCs两类码的最优性、组合编制及其相关问题展开了系统、深入的研究，取得了实质性的研究进展。研究成果包括：成功地导出了MLOOCs在一般参数条件下最优的上界，给出了若干特殊参数条件下最优的上界，创新地引入了RGDM的概念和其他构造方法，建立了大量多长度光正交码的存在性结果。项目组系统研究并建立了几类最优ESWC的存在性，并成功地给出了最优ESWC的渐近存在性。项目实施的四年间，项目组发表相关学术论文13篇，学术专著1部；其中 1 篇发表信息论顶级期刊《IEEE Trans. IT》上，2篇发表在《finite Fields and Their Applications》，1篇发表在《Journal of Combinatorial Designs》等专业权威期刊上。