正交拉丁超方的相关问题研究

正交拉丁k维超方是正交表的自然推广，是组合设计理论的重要研究内容之一，其在统计学、工程学、编码理论等领域中有着重要应用。正交拉丁k维超方有两种定义方式：一是维数意义上的正交，二是按照点对出现的次数相等。对于这两种定义的正交拉丁k维超方研究的核心问题为确定它们的最大个数以及构造问题。本项目的研究内容紧紧围绕其核心问题展开，具体包括：(1) 给出正交拉丁k维超方个数新的上下界；(2) 利用组合设计相关理论给出新的具体有效的构造方法和新结果。本项目的实施能够极大地丰富组合设计理论的内容，不仅有理论价值，而且有着重要的应用背景和实际意义。

Orthogonal Latin k-dimensional hypercubes are a natural generalization of orthogonal arrays and are also one of the important research topics in design theory, and have important applications in statistics, engineering, coding theory etc. There are two ways to define their orthogonality: one is dimensionally orthogonal and the other is according to the equal number of occurrences of each ordered pairs of symbols. For these two definitions of orthogonal Latin k-dimensional hypercubes, the core problem is to determine the maximum number of a set of orthogonal Latin k-dimensional hypercubes and to construct them. The research content of this project is based on these important problems, including: (1) give the new upper and lower bounds for a set of orthogonal Latin k-dimensional hypercubes; (2) give some new concrete and effective construction methods and results by using the theory of combinatorial designs. The implementation of this project can greatly enrich the content of combinatorial design theory. It not only has its own theoretical value, but also has important application background and practical significance.