| Abstract: | Let n and k be integers, with  and  . An  semi‐Latin square S is an  array, whose entries are k‐subsets of an  ‐set, the set of symbols of S, such that each symbol of S is in exactly one entry in each row and exactly one entry in each column of S. Semi‐Latin squares form an interesting class of combinatorial objects which are useful in the design of comparative experiments. We say that an  semi‐Latin square S is uniform if there is a constant μ such that any two entries of S, not in the same row or column, intersect in exactly μ symbols (in which case  ). We prove that a uniform  semi‐Latin square is Schur‐optimal in the class of  semi‐Latin squares, and so is optimal (for use as an experimental design) with respect to a very wide range of statistical optimality criteria. We give a simple construction to make an  semi‐Latin square S from a transitive permutation group G of degree n and order  , and show how certain properties of S can be determined from permutation group properties of G. If G is 2‐transitive then S is uniform, and this provides us with Schur‐optimal semi‐Latin squares for many values of n and k for which optimal  semi‐Latin squares were previously unknown for any optimality criterion. The existence of a uniform  semi‐Latin square for all integers  is shown to be equivalent to the existence of  mutually orthogonal Latin squares (MOLS) of order n. Although there are not even two MOLS of order 6, we construct uniform, and hence Schur‐optimal,  semi‐Latin squares for all integers  . & 2012 Wiley Periodicals, Inc. J. Combin. Designs 00: 1–13, 2012 |
