Combinatorial methods in networks and communications

该提案是我目前研究工作的延续，由NSERC发现基金支持。在网络和通信领域有很多研究课题都与组合数学有关。本提案的主要目的是对这些领域的一些重要课题进行研究。该方案的一个显著特点是组合方法是接近目标的主要工具。我以前工作的许多有趣和重要的结果表明，组合方法在这些领域是强大而有用的研究工具。然而，由于该研究需要网络、通信和组合学方面的知识，目前在这方面的研究人员还不够多。大多数研究组合学的研究人员都是从事数学工作，而不是从事网络和通信工作，而从事网络和通信工作的研究人员对组合学并不熟悉。本提案致力于通过研究这些领域的几个重要课题和培训该领域的高素质年轻研究人员来促进网络和通信代码的组合方法。本研究将招收博士后和硕士生。人们期待新的理论和有效的协议能够被提出来解决这些领域的重要问题。组合数学的理论也将得到进一步的发展。在通信码方面，本文重点研究了构建最优二维光正交码，这是支持共享介质中多用户同时使用的最重要技术之一，可以提高光纤的传输容量；构建传感器网络密钥分配方案，这是保证传感器网络通信安全的重要步骤；构建传感器网络调度方案，有效延长传感器网络的使用寿命。在网络、网络安全和通信编码的各种问题中，有一些具有特殊性质的集合系统，如循环和半循环设计、超简单循环设计、排列阵列以及其他一些组合设计和相关编码。本提案也将进一步探讨这些组合主题。我相信这项研究将为现实世界的应用创造重要的成果，并在组合数学和编码理论的基础理论方面取得成果。

This proposal is a continuation of my current research work which is supported by an NSERC Discovery Grant. There are a lot ofresearch topics in network and communication, which are related to combinatorial mathematics.The main purpose of this proposal is to do research on some important topics in these areas. One of the distinctive characteristics of this proposal is that combinatorial methods are the main tools to approach the destination. Many interesting and important results of my previous work show that combinatorial methods are powerful and useful research tools in these areas. However, since knowledge in networks, communication and combinatorics are required in this research, not enough researchers are now working in this direction. Most researchers involved in combinatorics are working in mathematics but not in networks and communications, while researchers working in networks and communications are not familiar with combinatorics. This proposal is dedicated to promoting combinatorial methods in networks and communication codes by investigating several important topics in these fields and by training highly qualified young researchers in this area. Post-doctoral fellows and master students will be involved in this research. It is expected that new theories and efficient protocols will be proposed to solve important problems in these fields. And theories of combinatorial mathematics will be further developed as well. Regarding communication codes, this proposal focuses on several topics including constructing optimal 2-dimensional optical orthogonal codes which is one of the mostimportant techniques supporting many simultaneous users in shared media so as to increase the transmission capacity of an optical fibre, constructing sensor network key distribution schemes which is an important step to secure sensor network communications, and constructing sensor network schedule schemes which can be used toprolong the lifetime of sensor networks effectively.There are several set systems with special properties, which are related tovarious problems in network, network security and communication codes, such as cyclic and semi-cyclic designs, super-simple cyclic designs, permutation arrays, and some other combinatorial designs and related codes. This proposal will further explore these combinatorial topics as well. I believe the research will createimportant results for real world applications and results in fundamental theories in combinatorial mathematics and coding theory.