Uniform semi-Latin squares and their pairwise-variance aberrations
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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 nk-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 dual of) 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, and from the uniform (6× 6)/10 semi-Latin squares classified, we obtain many affine resolvable designs for 72 treatments in 36 blocks of size 12, as well as new balanced incomplete-block designs for 36 treatments in 84 blocks of size 6.
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