Araştırma Makalesi
BibTex RIS Kaynak Göster

Recent results on Choi's orthogonal Latin squares

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

Öz

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.

Kaynakça

  • [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.
Yıl 2022, , 17 - 27, 15.01.2022
https://doi.org/10.13069/jacodesmath.1056511

Öz

Kaynakça

  • [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.
Toplam 12 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Makaleler
Yazarlar

Jon-lark Kim Bu kişi benim 0000-0002-0517-9359

Dong Eun Ohk Bu kişi benim 0000-0002-7737-5199

Doo Young Park Bu kişi benim

Jae Woo Park Bu kişi benim 0000-0001-7404-0492

Yayımlanma Tarihi 15 Ocak 2022
Yayımlandığı Sayı Yıl 2022

Kaynak Göster

APA Kim, J.-l., Ohk, D. E., Park, D. Y., Park, J. W. (t.y.). 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, ve Jae Woo Park. “Recent Results on Choi’s Orthogonal Latin Squares”. Journal of Algebra Combinatorics Discrete Structures and Applications 9, sy. 1 t.y.: 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, ve J. W. Park, “Recent results on Choi’s orthogonal Latin squares”, Journal of Algebra Combinatorics Discrete Structures and Applications, c. 9, sy. 1, ss. 17–27, doi: 10.13069/jacodesmath.1056511.
ISNAD Kim, Jon-lark vd. “Recent Results on Choi’s Orthogonal Latin Squares”. Journal of Algebra Combinatorics Discrete Structures and Applications 9/1 (t.y.), 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 vd. “Recent Results on Choi’s Orthogonal Latin Squares”. Journal of Algebra Combinatorics Discrete Structures and Applications, c. 9, sy. 1, ss. 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.