现代科技中设计理论问题的研究

In resent, the research on optical code division multiple access (OCDMA) communication technology becomes a point of fiber-obtic communication technology for its numerous technical advantages. OCDMA is a most important techniques supporting many simultaneous users in shared media so as to increase the transmission capacity of an optical fiber. However, the sticking point of an OCDMA system is to choose spread spectrum sequences with good correlation properties. An optical orthogonal code is a family of sequences with good auto- and cross-correlation properties. Its study has been motivated by an application in an optical code-division multiple access system. Therefore, it is much suitable for a spread spectrum sequences as its good auto- and cross-correlation properties. Moreover, the optical orthogonal codes with large size and good correlation properties is equired for many modern technology, such as mobile communications frequency hopping, spread spectrum communications, radar, neural network and sonar signal design et cetera. Hence, many researchers work for the determination of maximum size and construction of an optical orthogonal code. Thereupon, at the first part of this project, we will continue preliminary work to determine the maximum size of two-dimensional and more dimensional optical orthogonal codes with different auto- and cross-correlation parameters, then give the construction of some infinite classes. . The graph coloring theory is a classical problems of the graph theory, but also one of the important research topic of combinatorial design theory. the fundamental discrete structure, so as the coloring of a hypergraph. A hypergraph is a partial design from perspective of combinatorial design. Hence, in second part of this project, we will dill with the unresolved coloring problem of hypergraphs by incessent interation method of figure valued function and homomorphic method, such as the coloring of block designs, direct product of hypergraphs, fractional coloring of hypergraphs et cetera. The study on hypergraph coloring will be necessary preparatory for our research of next steps. The hypergraph coloring theory is still with strong application background. It is applied to energy supplying problem, work scheduling problem and also the field of parallel computing, database management, and molecular biology et cetera.. It is well known for us that whether the size determination and construction of an optical orthogoanl code or hypergraph coloring are both theoritical research, but they have both academic significance and a wid range of application background.

近年来，光纤码分多址通信技术因其纵多技术优势成为光纤通信技术方面研究的一个热点. 然而，被用作扩频序列的光正交码的容量及构造问题也成为人们的研究热题. 在许多实际应用中需要相关性能好、多码字的光正交码. 于是，本项目第一部分，继续前期工作，拟确定自相关值和互相关值不同的二维及更多维光正交码的最大容量及构造问题，试构造出相关性能好且容量更大的光正交码；染色问题是图论中的经典问题之一，同时也是组合设计理论中的一项重要研究课题. 从组合设计的观点来看，一个超图是一个部分设计. 超图的染色问题也是近年来国内外研究的一个热点问题，而目前国内外研究者主要用算法分析方法研究图的染色问题. 在第二部分，本项目试将结合图值函数的无穷次迭代和同态法用于较大规模图类的染色问题，解决一些未解决的超图的染色问题，如，超图直积的染色，超图的分数染色，多维超图的染色，设计的染色等, 为以后的研究作前期准备.

光正交码是为码分多址光纤系统而设计的专用码. 而实际应用所需的是相关性能好的大容量的光正交码. 而获得大容量码字常用方法有两种：延长码长或增加维数. 光正交码的大量研究结果表明二维光正交码在一定程度上克服了一维光正交码存在的可用码字少、码长大、自相关性差等缺陷. 除了自相关值和互相关值都等于1的二维光正交码 , 自相关值为2，互相关值为1的二维光正交码 的相关性能是最好的.然而，确定这一类二维光正交码的最大容量和给出码字结构是我们理论研究的意义所在... 我们着力研究了二维光正交码 的最大容量和码子结构.对任意正整 和奇数 ，确定了 的最大容量的较紧的上界，并给出部分 的码字结构；对任意正整数 确定了 的最大容量的较紧的上界.经过直接构造和编程验证，我们所得到的理论上界确实紧的.至今，国内外关于二维光正交码 的最好的结果是王小苗和冯弢等确定了 的最大容量的上界. 我们的研究进一步推进了自相关值和互相关值不同的二维光正交码的研究工作.在光纤通讯技术中，这就相当于获得了更多相关性能较好的光码分多址系统的扩频序列，在推动通讯技术发展是有较大的理论意义. 除此之外，光正交码在移动通信，跳频，扩频通信，雷达，声纳信号设计和神经网络等方面也有众多的应用. 二维光正交码至使非相干光纤码分多址系统又呈现了良好的应用前景..染色问题是图论的一个经典问题，同时也是组合设计理论中的一个重要研究课题. Hedeteniemi染色猜想和平面图全染色猜想是至今未被攻克的数学难题，因此猜想在特殊图类方面的研究是很有意义的课题．. 我们根据极大扩容图的代数结构性质及与原图的关系，证明了简单图的若干次扩容图满足Hedeteniemi染色猜想，得到了对 Hedeteniemi染色猜想成立的无限类图. 还证明了最大度为7且3-圈和5-圈不正常相交的平面图的全色数等于8.这些研究对解决Hedeteniemi猜想和平面图全染色猜想有一定的参考价值和推进作用. 从而进一步肯定了该猜想的正确性. 染色问题在能源供应问题，工作排序问题,平行计算、数据库管理及分子生物学等领域都有广泛的应用.