正交频率方的构造及其应用

Although Latin squares have many useful properties, for some statistical applications these structures are too restrictive. The more general concepts of frequency squares and orthogonal frequency squares offer more flexibility. This item studies relationship between orthogonal frequency squares and orthogonal decomposition of projection matrix, relationship between orthogonal frequency squares and orthogonal array from tensor of matrix, transversal design, incomplete orthogonal Latin square, the contractive replacement method and the expansive replacement method, matrix image of orthogonal frequency squares, generalized difference matrix, generalized Hadamard product, generalized Kronecker sum, etc. And it will apply the methods to construct orthogonal array to obtain orthogonal frequency squares. It will set up a complete system for the construction of orthogonal mixed level frequency squares, in which there be addition, subtraction, multiplication, division of orthogonal frequency squares. This item will extend the method for only constructing one class of orthogonal frequency squares to that for constructing series of frequency squares. We will improve the lower bounds to obtain more new orthogonal frequency squares. By using relationship between orthogonal frequency squares and other design, it will study the properties of orthogonal frequency squares which will be used for construction of uniform design array, coding and other combinatorial structure. This item will enrich construction theory of orthogonal frequency squares and make orthogonal frequency squares have wider applications. Therefore, it will play an important role in theory and applications in experiment design.

尽管正交拉丁方有很多有用性质，但在统计应用上，其结构有较多的局限性，而正交频率方比正交拉丁方提供了更多的灵活性。本项目研究正交频率方和投影矩阵正交分解之间的关系，通过矩阵的拉长，建立正交表和正交频率方的联系，研究频率方和截态、正交带洞拉丁方等设计理论，扩张性和压缩性替换方法，正交频率方的矩阵象，广义差集矩阵、Hadamard矩阵的概念以及矩阵的Keronecker积、Hadamard积等运算，将我们提出的构造正交表的方法用于频率方、正交频率方的构造上，研究频率方的加减乘除法，建立一套较完整的正交频率方构造方法和体系。研究和改进正交频率方的下界，以获得更加丰富的新的正交频率方。研究频率方和编码之间的关系，将频率方应用于编码、网格、横街设计等组合结构的构造上，该项目将使正交频率方的构造理论更加丰富，使正交频率方的应用前景将更为广阔。

尽管正交拉丁方有很多有用性质，但在统计应用上，其结构有较多的局限性，而正交频率方比正交拉丁方提供了更多的灵活性。本项目研究正交频率方和投影矩阵正交分解之间的关系，通过矩阵的拉长，建立正交表和正交频率方的联系，定义了频率立方、频率超方、正交频率立方、正交频率超方、量子频率方、量子频率立方、量子频率超方、正交量子频率方、正交量子频率立方、正交量子频率超方等组合设计和量子组合设计。一方面，通过扩张性和压缩性替换方法，Hamming距离，差集矩阵以及Keronecker积、Hadamard积等运算，利用构造正交表的方法研究了这些组合设计的加减乘除法，建立一套较完整的正交频率方构造方法和体系。另一方面，通过非冗余正交表，研究正交量子频率方、正交量子频率立方、正交量子频率超方和量子纠缠态之间的关系。本项目将Hamming距离应用于量子纠缠态的构造当中，采用高强度正交表的构造方法解决了2-和3-级均匀态存在性和构造方面的公开问题，这是量子理论和离散数学交叉领域著名的难问题，在量子信息理论方面具有潜在的影响。通过弹性函数的支撑矩阵应用到密码学上，彻底解决了一类旋转对称弹性函数的构造和计数问题。该项目将使正交频率方的构造理论更加丰富，使正交频率方的应用前景将更为广阔。