Recent results on Choi's orthogonal Latin squares

Year 2022, Volume: 9 Issue: 1, 17 - 27, 15.01.2022

Jon-lark Kim Dong Eun Ohk Doo Young Park Jae Woo Park

https://doi.org/10.13069/jacodesmath.1056511

Abstract

Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler although this was less known. He introduced a pair of orthogonal Latin squares of order 9 in his book. Interestingly, his two orthogonal non-double-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi's Latin squares of order 9. The most recent one is Ko-Wei Lih's construction of Choi's Latin squares of order 9 based on the two $3 \times 3$ orthogonal Latin squares.
In this paper, we give a new generalization of Choi's orthogonal Latin squares of order 9 to orthogonal Latin squares of size $n^2$ using the Kronecker product including Lih's construction. We find a geometric description of Choi's orthogonal Latin squares of order 9 using the dihedral group $D_8$. We also give a new way to construct magic squares from two orthogonal non-double-diagonal Latin squares, which explains why Choi's Latin squares produce a magic square of order 9.

Keywords

Choi Seok-Jeong, Koo-Soo-Ryak, Latin squares, Magic squares

References

• [1] J. W. Brown, F. Cherry, L. Most, M. Most, E. T. Parker, W. D. Wallis, Completion of the spectrum of orthogonal diagonal Latin squares, Graphs, Matrices and Desings, Dekker (1993) 43–49.
• [2] S. J. Choi, Gusuryak, Seoul National University Kyujanggak Institute for Korean Studies.
• [3] C. J. Colbourn, J. H. Dinitz, Handbook of combinatorial designs, CRC Press, Second Edition (2007).
• [4] L. Euler, De Quadratis Magicis, Commentationes Arithmeticae Collectae 2 (1849) 593-602 and Opera Omnia 7 (1911) 441–457.
• [5] M. A. Francel , D. J. John, The dihedral group as the array stabilizer of an augmented set of mutually orthogonal Latin squares, Ars Combin. 97 (2010) 235–252.
• [6] A. J. W. Hilton, Some simple constructions for double diagonal Latin squares, Sankhya: The Indian Journal of Statistics 36(3) (1974) 215–229.
• [7] A. J. W. Hilton, S. H. Scott, A further construction of double diagonal orthogonal Latin squares, Discrete Mathematics 7 (1974) 111–127.
• [8] A. D. Keedwell, J. DÃlnes, Latin squares and their applications, Academic Press, Second Edition (2015).
• [9] C. F. Laywine, G. L. Mullen, Discrete mathematics using Latin squares, John Wiley & Sons, New York (1998).
• [10] K. W. Lih, A remarkable Euler square before Euler, Mathematics Magazine 83(3) (2010) 163–167.
• [11] H. Y. Song, Choi’s orthogonal Latin squares is at least 67 years earlier than Euler’s, Global KMS Conference, Jeju, Korea (2008).
• [12] Y. Zhang, K. Chen, N. Cao, H. Zhang, Strongly symmetric self-orthogonal diagonal Latin squares and Yang Hui type magic squares, Discrete Mathematics 328 (2014) 79–87.