Optimal and efficient semi-Latin squares

Let n and k be integers, with n > 1 and k > 0. An (n × n)/k semi-Latin square S is an n × n array, whose entries (called blocks) are k-element subsets of a set of size nk, the set of symbols of S, such that each symbol of S occurs exactly once in each row and exactly once in each column of S. Semi-Latin squares form a class of 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 equireplicate block designs are presented. Applications include the complete enumeration of the (4× 4)/k semi-Latin squares for k ≤ 10, and the determination of those that are A-, Dand 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. [