Acknowledgement
The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (NRF-2017R1D1A3B05029924).
References
- R. Baer, Nets and groups, Trans. Amer. Math. Soc. 46 (1939), 110-141. https://doi.org/10.2307/1989995
- P. J. Cameron, Metric and geometric properties of sets of permutations, in Algebraic, extremal and metric combinatorics, 1986 (Montreal, PQ, 1986), 39-53, London Math. Soc. Lecture Note Ser., 131, Cambridge Univ. Press, Cambridge, 1988.
- M. B. Cohen, C. J. Colbourn, and A. C. H. Ling, Constructing strength three covering arrays with augmented annealing, Discrete Math. 308 (2008), no. 13, 2709-2722. https://doi.org/10.1016/j.disc.2006.06.036
- W. Glunt, T. L. Hayden, and R. Reams, The nearest generalized doubly stochastic matrix to a real matrix with the same first and second moments, Comput. Appl. Math. 27 (2008), no. 2, 201-210. https://doi.org/10.1590/S0101-82052008000200005
- B. Im and J. D. H. Smith, Orthogonality of approximate Latin squares and quasigroups, in Nonassociative Mathematics and its Applications, P. Vojtechovsky et al. Eds., Contemporary Mathematics 721, 165-181, American Mathematical Society, Providence, RI, 2019.
- C. W. H. Lam, L. Thiel, and S. Swiercz, The nonexistence of finite projective planes of order 10, Canad. J. Math. 41 (1989), no. 6, 1117-1123. https://doi.org/10.4153/CJM1989-049-4
- H.-Y. Lee, B. Im, and J. D. H. Smith, Stochastic tensors and approximate symmetry, Discrete Math. 340 (2017), no. 6, 1335-1350. https://doi.org/10.1016/j.disc.2017.02.013
- M. E. O'Nan, Sharply 2-transitive sets of permutations, in Proceedings of the Rutgers group theory year, 1983-1984 (New Brunswick, N.J., 1983-1984), 63-67, Cambridge Univ. Press, Cambridge, 1985.
- J. D. H. Smith, An Introduction to Quasigroups and Their Representations, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2007.
- J. D. H. Smith, Introduction to Abstract Algebra, 2nd. ed., Chapman and Hall/CRC, Boca Raton, FL, 2016.
- J. D. H. Smith and A. B. Romanowska, Post-Modern Algebra, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. https://doi.org/10.1002/9781118032589
- A. Smoktunowicz, R. Kozera, and G. Oderda, Efficient numerical algorithms for constructing orthogonal generalized doubly stochastic matrices, Appl. Numer. Math. 142 (2019), 16-27. https://doi.org/10.1016/j.apnum.2019.02.008