Recent results on Choi's orthogonal Latin squares

Year 2022, , 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

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