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Recent results on Choi's orthogonal Latin squares

Year 2022, , 17 - 27, 15.01.2022
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.

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.
Year 2022, , 17 - 27, 15.01.2022
https://doi.org/10.13069/jacodesmath.1056511

Abstract

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.
There are 12 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Jon-lark Kim This is me 0000-0002-0517-9359

Dong Eun Ohk This is me 0000-0002-7737-5199

Doo Young Park This is me

Jae Woo Park This is me 0000-0001-7404-0492

Publication Date January 15, 2022
Published in Issue Year 2022

Cite

APA Kim, J.-l., Ohk, D. E., Park, D. Y., Park, J. W. (n.d.). Recent results on Choi’s orthogonal Latin squares. Journal of Algebra Combinatorics Discrete Structures and Applications, 9(1), 17-27. https://doi.org/10.13069/jacodesmath.1056511
AMA Kim Jl, Ohk DE, Park DY, Park JW. Recent results on Choi’s orthogonal Latin squares. Journal of Algebra Combinatorics Discrete Structures and Applications. 9(1):17-27. doi:10.13069/jacodesmath.1056511
Chicago Kim, Jon-lark, Dong Eun Ohk, Doo Young Park, and Jae Woo Park. “Recent Results on Choi’s Orthogonal Latin Squares”. Journal of Algebra Combinatorics Discrete Structures and Applications 9, no. 1 n.d.: 17-27. https://doi.org/10.13069/jacodesmath.1056511.
EndNote Kim J-l, Ohk DE, Park DY, Park JW Recent results on Choi’s orthogonal Latin squares. Journal of Algebra Combinatorics Discrete Structures and Applications 9 1 17–27.
IEEE J.-l. Kim, D. E. Ohk, D. Y. Park, and J. W. Park, “Recent results on Choi’s orthogonal Latin squares”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 9, no. 1, pp. 17–27, doi: 10.13069/jacodesmath.1056511.
ISNAD Kim, Jon-lark et al. “Recent Results on Choi’s Orthogonal Latin Squares”. Journal of Algebra Combinatorics Discrete Structures and Applications 9/1 (n.d.), 17-27. https://doi.org/10.13069/jacodesmath.1056511.
JAMA Kim J-l, Ohk DE, Park DY, Park JW. Recent results on Choi’s orthogonal Latin squares. Journal of Algebra Combinatorics Discrete Structures and Applications.;9:17–27.
MLA Kim, Jon-lark et al. “Recent Results on Choi’s Orthogonal Latin Squares”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 9, no. 1, pp. 17-27, doi:10.13069/jacodesmath.1056511.
Vancouver Kim J-l, Ohk DE, Park DY, Park JW. Recent results on Choi’s orthogonal Latin squares. Journal of Algebra Combinatorics Discrete Structures and Applications. 9(1):17-2.