基于正交表大集和Langford序列的几类幻表的构造

The integer matrices with the magic sum property such as multimagic square, magic square with all subsquares of possible orders, Yang Hui type magic square, addition-multiplication magic square, signed magic array, multidimensional magic rectangle, sparse anti-magic square have been one of the hot research questions in combinatorial designs in recent years. The research of the construction methods and the existence on these magic arrays have not only strong theoretical significance in combinatorial designs, but also practical value in graph theory and information science. Large sets of orthogonal arrays and Langford sequences and their generalized form can be used to construct various types of magic arrays. In this project, we will study various types of generalized forms of large sets of orthogonal arrays and Langford sequences and related combinatorial designs, and use them to construct various types of magic arrays. The followings are due to be obtained. (1) large sets of orthogonal arrays are used to get some new families of multimagic squares; (2) use extended Langford sequences to completely solve the existence of magic squares with all subsquares of possible orders; (3) generalize large sets of orthogonal arrays to get new constructions of Yang Hui type magic squares; (4) establish the relationship between large sets of orthogonal arrays and addition-multiplication magic squares; (5) completely solve the existence of signed magic arrays with the property that the column number is the multiple of row number; (6) completely solve the existence of sparse anti-magic square based on Langford sequences, and some magic rectangles are due to be obtained. Research the related designs and the applications. The research results will enrich the theory of combinatorial design and promote the development of relevant applications.

多重幻方、全子幻方、杨辉型幻方、加乘幻方、符号幻表、多维幻矩、稀疏反幻方等具有幻和性质的整数矩阵是近几年组合设计研究的热点之一。研究这些幻表的构造方法及存在性，不仅具有较强的组合设计理论意义，同时在图论、信息科学中有一定的实用价值。正交表大集和Langford序列及其推广形式可用来构造各种类型的幻表。本项目拟研究正交表大集和Langford序列的各种推广形式和与之相关联的组合设计，并用来构造各种类型的幻表。拟得到如下成果：（1）利用正交表强双大集得到多重幻方的新的类；（2）利用Langford序列完整解决全子幻方的存在性；（3）推广正交表大集以得到杨辉型幻方的新构造；（4）建立基于正交表大集的加乘幻方的构造；（5）完整解决列数是行数倍数的符号幻表的存在性；（6）利用Langford序列完整解决稀疏反幻方的存在性，并得到几类多维幻矩。研究成果将丰富组合设计理论并推动相关应用发展。

幻表是一类具有特定幻和性质的整数矩阵，包括各种类型的幻方、幻矩、符号幻表、稀疏反幻方等。对其构造方法及存在性的研究，不仅具有较强的组合设计理论意义，同时在图论、信息科学等领域有一定的实用价值。本项目应用正交表大集、Langford序列和与之相关联的组合设计构造各种类型的幻表，按照计划完成了以下研究工作。.系统研究了t-重幻方的组合构造，包括：正交表双大集和t-重幻矩、t-重幻方、泛对角t-重幻方、杨辉型t-重幻方等。利用正交表强双大集给出了多重幻方新的乘积构造，利用频率方得到了强对称弱泛对角自正交对角拉丁方的一个构造。.研究了全子幻方，利用扩展的Lanford序列，给出了一类奇数阶全子幻方的构造和存在性证明，其方法可用于完整解决全子幻方的存在性。.研究了平方幻方，利用正交对角拉丁方、互补对、广义行平方幻矩和幻对等，得到了偶数阶正规平方幻方的谱。.研究了稀疏反幻方的存在性，提出了均匀正则稀疏矩阵和伪稀疏反幻方的概念，完整解决了奇数阶正则稀疏反幻方的存在性，并得到了相关图标号的结论。.研究了非冗余正交表的若干构造并且得到一些新结果，得到阶数为v的3次幂的两类非冗余正交表，利用差阵的构造方法得到了强度为3的阶数是非素数幂的Augmented正交表的无穷类，得到了量子纠缠中的一些类型的k-均匀态。.研究了基于各类幻表、Langford序列、混沌序列等各类组合设计的图像加密方法。

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