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。

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.