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The Existence and Application of Strongly Idempotent Self-orthogonal Row Latin Magic Arrays

The Existence and Application of Strongly Idempotent Self-orthogonal Row Latin Magic Arrays 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

The Existence and Application of Strongly Idempotent Self-orthogonal Row Latin Magic Arrays

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References (11)

Publisher
Springer Journals
Copyright
Copyright © 2018 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-018-0777-3
Publisher site
See Article on Publisher Site

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.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Oct 4, 2018

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