Automating the construction and evaluation of eighteen-treatment semi-Latin squares of order six in blocks of size three

A Microsoft Excel program that has the capacity of constructing  semi-Latin squares from  reduced Latin squares and the seven distinct   semi-Latin squares (SLS)of Bailey and Royle (1997) is here presented. The aim is to facilitate, through the knowledge of computer programming, the construction and statistical evaluation of the eighteen-treatment semi-Latin squares of order six and block size three. The eighteen-treatment semi-Latin squares of order six in blocks of size three are constructed by applying the method of superposition. The program also computes and displays the incidence matrices of all the constructed SLS as well as the, A-, D-, E- and MV-efficiency criteria for evaluating the usefulness of the design for experimentation. The program has a size of 20 kilobytes on hard disk space and takes about thirty minutes of computing time to construct and evaluate the semi-Latin squares. Illustrative examples were given to demonstrate the performance of the program.

Bailey, R.A. (1988). Semi-Latin Squares, Journal of Statistical Planning and Inference, 18, 299 – 312.

Bailey, R.A. (2000). Semi-Latin Squares: Constructions. Available at http://www.maths.qmul.ac.uk/~rab/slsconst.html.

Bailey, R.A.(1992), Efficient Semi-Latin Squares, Statistica Sinica, 2, 413–437.

Bailey, R.A. and Royle, G. (1997). Optimal Semi-Latin Squares with Side Six and Block Size Two, Proceedings of the Royal Society, 453 (1964), 1904 – 1914.

Cheng, C.-S. and Bailey, R.A.(1991). Optimality of some Two-Associate-Class Partially Balanced Incomplete-Block Designs, Annals of Statistics, 19, 1667–1671.

Chigbu, P.E. and Eze, B.C. (2001). Automating the Group-Theoretic-Based Construction Procedure for the ((n×n))⁄k Semi-Latin Square, Utilitas Mathematica, 60, 107 – 123.

Mba, E.I., Chigbu, P.E. and Ukaegbu, E.C. (2021).An Efficient Semi-Latin Square with Side Six and Block Size Three, Preprint.

McKay, B.D. (1981). Practical Graph Isomorphism, Congressus Numerantium, 30 (1981), 45 – 87.

McKay, B.D.  and Piperno, A. (2014). Practical Graph Isomorphism II, Journal of Symbolic Computation, 60, 94 – 112.

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

Soicher, L.H. (1993). GRAPE: A System for Computing with Graphs and Groups, In: Larry Finkelstein and William M. Kantor(Editors), Groups and Computation, Vol. 11 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 287–291, American Mathematical Society.

Soicher, L.H. (2013a). Optimal and Efficient Semi-Latin Squares, Journal of Statistical Planning and Inference, Vol. 143, 573 – 582.

Soicher, L.H. (2013b). Designs, Groups and Computing, In: Detinko, Flanery and O’Brien (Editors), Probabilistic Group Theory, Combinatorics and Computing: Lecture Notes from Fifth de Brun Workshop, Springer-Verlag, London.