Uniform semi-Latin squares and their pairwise-variance aberrations

Abstract

For integers $n>2$ and $k>0$, an $(n\times n)/k$ semi-Latin square is an $n\times n$ 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. We show that when a uniform $(n\times n)/k$ semi-Latin square exists, the Schur optimal $(n\times n)/k$ 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 $(n\times n)/k$ semi-Latin squares with minimum PV aberration when there exist $n-1$ mutually orthogonal Latin squares (MOLS) of order $n$. These do not exist when $n=6$, and the smallest uniform semi-Latin squares in this case have size $(6\times 6)/10$. We present a complete classification of the uniform $(6\times 6)/10$ semi-Latin squares, and display the one with least PV aberration. We give a construction producing a uniform $((n+1)\times (n+1))/((n-2)n)$ semi-Latin square when there exist $n-1$ MOLS 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 (BIBDs) can be constructed from uniform semi-Latin squares. From the uniform $(6\times 6)/10$ 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 BIBDs for 36 treatments in 84 blocks of size 6. In particular, this shows that there are at least 16875 pairwise non-isomorphic orthogonal arrays $\mathrm{OA}(72,6,6,2)$.