Optimal and efficient semi-Latin squares

An ( n × n ) / k semi-Latin square is an n×n square array in which nk distinct symbols (representing treatments) are placed in such a way that there are exactly k symbols in each cell (row–column intersection) and each symbol occurs once in each row and once in each column. Semi-Latin squares form a class of row–column designs generalising Latin squares, and have applications in areas including the design of agricultural experiments, consumer testing, and via their duals, human–machine interaction. In the present paper, new theoretical and computational methods are developed to determine optimal or efficient ( n × n ) / k semi-Latin squares for values of n and k for which such semi-Latin squares were previously unknown. The concept of subsquares of uniform semi-Latin squares is studied, new applications of the DESIGN package for GAP are developed, and exact algebraic computational techniques for comparing efficiency measures of binary equireplicate block designs are described. Applications include the complete enumeration of the ( 4 × 4 ) / k semi-Latin squares for k = 2 , … , 10 , and the determination of those that are A-, D-, and E-optimal, the construction of efficient ( 6 × 6 ) / k semi-Latin squares for k = 4 , 5 , 6 , and counterexamples to a long-standing conjecture of R.A. Bailey and to a similar conjecture of D. Bedford and R.M. Whitaker.