Optimal and efficient eighteen-treatment semi-Latin squares in blocks of size three

Semi-Latin squares with side six and block size three, constructed by superimposing each of the 9408 reduced Latin squares of order six on each of certain ((6×6))⁄2 semi-Latin squares, are here presented. The aim was to identify optimal and/or efficient semi-Latin square(s) of order six from the 9408 reduced Latin squares of order six. A Microsoft Excel program was used to facilitate the construction by superposition and the statistical evaluation of the corresponding semi-Latin squares of sides six in blocks of size three by computing their A, D, E and MV statistical efficiency measures A total of 65856 semi-Latin squares with side six and blocks of size three were constructed and evaluated. One of the semi-Latin squares was identified to be A-optimal, D-optimal, E-optimal and MV-optimal. Also, with respect to the efficiency measures, the same optimal semi-Latin square is the most efficient of the 65856 semi-Latin squares. This efficient semi-Latin square which has canonical efficiency factors, 0.5980 with multiplicity three, 0.6667 with multiplicity ten, 0.8464 with multiplicity three and 1.0 with multiplicity one, is a simple orthogonal multi-array (SOMA) of order six; specifically denoted by SOMA(3, 6). Also, this optimal and efficient semi-Latin square has the same A, D, E and MV statistical efficiency values with the two indecomposable SOMA(3, 6) developed by Soicher as well as the efficient semi-Latin square of order six by Bailey.

Bailey, R.A. (1992). Efficient semi-Latin squares, Statistica Sinica, 2, 413-437.

Bailey, R.A. (1997). A Howell design admitting A₅¹, Discrete Mathematics, 167/168, 65 – 71.

Bailey, R.A. (2011). Symmetric factorial designs in blocks, Journal of Statistical Theory and Practice, 5(1), 13 – 24.

Bailey, R.A. and Chigbu, P.E. (1997). Enumeration of semi-Latin squares, Discrete Mathematics, 167/168, 73 – 84.

Bailey, R.A. and Royle, G. (1997). Optimal semi-Latin squares with side six and block size two, Proceedings of the Royal Society London A, 453 (1964), 1903 – 1914.

Bailey, R.A. (1990). An Efficient semi-Latin square for twelve treatments in blocks of size two, Journal of Statistical Planning and Inference, 26, 263-266.

Chigbu, P.E., Mba, E.I. and Ukaegbu, E.C. (2021). Automating the construction and evaluation of eighteen-treatment semi-Latin squares of order six in blocks of size three. Journal of the Nigerian Statistical Association, 33, 80 – 99.

Denes, J. and Keedwell, A.D. (1991). Latin squares - new developments in the theory and applications, Annals of Discrete Mathematics, 46, 1 – 453.

Hung, S.H.Y. and Mendelsohn, N.S. (1974). On Howell designs, Journal of Combinatorial Theory, A 16, 174 – 198.

Kreher, D.L., Royle, G.F. and Wallis, W.D. (1996). A family of resolvable regular graph designs, Discrete Mathematics, 156, 269 – 275.

McKay, B.D. (2019). Reduced Latin squares, McKay’s Personal webpage. Available at www.users.cecs.anu.au/~bdm/.

McKay, B.D. and Wanless, I.M. (2005). On the number of Latin squares. Annals of Combinatorics, 9, 335 – 344.

Meszka, M. & Tyniec, M. (2019). Orthogonal one-factorizations of complete multipartite graphs, Designs, Codes and Cryptography, 87, 987 – 993.

Phillips, N.C.K. and Wallis, W.D. (1996). All solutions to a tournament problem, Congressum Numerantium, 114, 193 – 196.

Preece, D.A. and Philips, N.C.K. (2002). Euler at the bowling green, Utilitas Mathematica, 61, 129 – 165.

Seah, E. and Stinson, D.R. (1987). An assortment of new Howell designs, Utilitas Mathematica, 31, 175 – 188.

Soicher, L.H. (2013). Optimal and efficient semi-Latin squares, Journal of Statistical Planning and Inference, 143, 573 – 582.

Soicher, L.H. (1999). On the structure and classification of SOMAs: Generalizations of mutually orthogonal Latin squares. Electronic Journal of Combinatorics, 6 (1), R32. Available at www.elibm.org/article/10002268.

Soicher, L.H. (2012a). SOMA Update, http://www.maths.qmul.ac.uk/~leonard/soma.

Soicher, L.H. (2012b). Uniform semi-Latin squares and their Schur-optimality, Journal of Combinatorial Design, 20, 265 – 277.