The Existence and Application of Strongly Idempotent Self-orthogonal Row Latin Magic Arrays |
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Authors: | Yu-fang Zhang Jing-yuan Chen Dian-hua Wu Han-tao Zhang |
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Institution: | 1.Department of Mathematics, Guangxi,Normal University,Guilin,China;2.School of Mathematics and Statistics,Xinyang Normal University,Xinyang,China;3.Computer Science Department,The University of Iowa,Iowa City,USA |
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Abstract: | Let N = {0, 1, · · ·, n ? 1}. A strongly idempotent self-orthogonal row Latin magic array of order n (SISORLMA(n) for short) based on N is an n × n array M satisfying the following properties: (1) each row of M is a permutation of N, and at least one column is not a permutation of N; (2) the sums of the n numbers in every row and every column are the same; (3) M is orthogonal to its transpose; (4) the main diagonal and the back diagonal of M are 0, 1, · · ·, n ? 1 from left to right. In this paper, it is proved that an SISORLMA(n) exists if and only if n ? {2, 3}. As an application, it is proved that a nonelementary rational diagonally ordered magic square exists if and only if n ? {2, 3}, and a rational diagonally ordered magic square exists if and only if n ≠ 2. |
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