Uniform semi-Latin squares and their pairwise-variance aberrations

Abstract For integers n > 2 and k > 0 , an ( n × n ) ∕ k semi-Latin square is an n × n array of k -subsets (called blocks) of an n k -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 ( n × n ) ∕ k semi-Latin square is Schur optimal in the class of all ( n × n ) ∕ k semi-Latin squares, and here we show that when a uniform ( n × n ) ∕ k semi-Latin square exists, the Schur optimal ( n × n ) ∕ k 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 ( n × n ) ∕ k semi-Latin squares with minimum PV aberration when there exist n − 1 mutually orthogonal Latin squares of order n . These do not exist when n = 6 , and the smallest uniform semi-Latin squares in this case have size ( 6 × 6 ) ∕ 10 . We present a complete classification of the uniform ( 6 × 6 ) ∕ 10 semi-Latin squares, and display the one with least PV aberration. We give a construction producing a uniform ( ( n + 1 ) × ( n + 1 ) ) ∕ ( ( n − 2 ) n ) semi-Latin square when there exist n − 1 mutually orthogonal Latin squares of order n , 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 ( 6 × 6 ) ∕ 10 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 OA ( 72 , 6 , 6 , 2 ) .