等内积对角拉丁方正交偶及两类幻方  

Orthogonal Diagonal Latin Squares with Equiscalar Product and Two Kinds of Magic Squares

作　　者：张世德[1] 李程 张迪 ZHANG Shi-de;LI Cheng;ZHANG Di(College Mathematical and Information Science,Henan Normal University,Xinxiang 453007,China;Daxiang Press Co.Ltd.Zhengzhou 450044,China;Civil Safe Insurance Pic,Phnom Penh,Cambodia)

机构地区：[1]河南师范大学数学与信息科学学院,河南新乡453007 [2]大象出版社有限公司,河南郑州450044 [3]泰安保险公共有限公司,金边市柬埔寨

出　　处：《数学的实践与认识》2020年第24期215-230,共16页Mathematics in Practice and Theory

摘　　要：建立3^2阶和(2k+1)^2(3■(2k+1))阶等内积性对角拉丁方正交偶的一般公式,引入矩阵的函数型Kronecker积的概念,以此为基础解决(2m+1)^2(m≥1)阶等内积性对角拉丁方正交偶的存在性和构造性问题.进而,(2m+1)^2(m≥1)阶二次幻方、加乘幻方的存在性和构造性问题得到彻底解决,且将构成两类幻方的数集拓广至二维等差矩阵.For the cases of orders 3^2 and(2k+1)^2(3■(2k+1)),the general formulae for a pair of orthogonal diagonal Latin squares with the property of equi-scalar product have been established directly.We introduce a number of concepts of the functional Kronecker products which are used to obtain the existence and construction of orthogonal pair of diagonal Latin squares with the property of equi-scalar of order(2m+1)^2(m≥1).As an application,the existence and construction problems of quadratic magic squares and addition-multiplication magic squares of order(2m+1)^2 have been solved and the number sets constructing such kinds of magic squares can be extended to the arithmetic matrices in each direction.

关 键 词：等度方阵 矩阵的函数型Kronecker积 等内积对角拉丁方正交偶 二维等差矩阵 二次幻方 加乘幻方 

分 类 号：O151.21[理学—数学]