Uniform semi-Latin squares and their pairwise-variance aberrations

Abstract

For integers and , an semi-Latin square is an array of -subsets (called blocks) of an -set (of treatments), such that each treatment occurs once in each row and once in each column of the array. A semi-Latin square is uniform if every pair of blocks, not in the same row or column, intersect in the same positive number of treatments. It is known that a uniform semi-Latin square is Schur optimal in the class of all semi-Latin squares, and here we show that when a uniform semi-Latin square exists, the Schur optimal semi-Latin squares are precisely the uniform ones. We then compare uniform semi-Latin squares using the criterion of pairwise-variance (PV) aberration, introduced by J. P. Morgan for affine resolvable designs, and determine the uniform semi-Latin squares with minimum PV aberration when there exist mutually orthogonal Latin squares of order . These do not exist when , and the smallest uniform semi-Latin squares in this case have size . We present a complete classification of the uniform semi-Latin squares, and display the one with least PV aberration. We give a construction producing a uniform semi-Latin square when there exist mutually orthogonal Latin squares of order , and determine the PV aberration of such a uniform semi-Latin square. Finally, we describe how certain affine resolvable designs and balanced incomplete-block designs can be constructed from uniform semi-Latin squares. From the uniform semi-Latin squares we classified, we obtain (up to block design isomorphism) exactly 16875 affine resolvable designs for 72 treatments in 36 blocks of size 12 and 8615 balanced incomplete-block designs for 36 treatments in 84 blocks of size 6. In particular, this shows that there are at least 16875 pairwise non-isomorphic orthogonal arrays .