Uniform semi-Latin squares and their pairwise-variance aberrations

For integers n>2 and k>0, an (n×n)?k semi-Latin square is n×n array of k-subsets (called blocks) nk-set (of treatments), such that each treatment occurs once in row column the array. A uniform if every pair blocks, not same or column, intersect positive number treatments. It known a Schur optimal class all squares, here we show when exists, squares are precisely ones. We then compare using criterion pairwise-variance (PV) aberration, introduced by J. P. Morgan for affine resolvable designs, determine with minimum PV aberration there exist n?1 mutually orthogonal Latin order n. These do n=6, smallest this case have size (6×6)?10. present complete classification (6×6)?10 display one least aberration. give construction producing ((n+1)×(n+1))?((n?2)n) n, square. Finally, describe how certain designs balanced incomplete-block can be constructed from squares. From classified, obtain (up to block design isomorphism) exactly 16875 72 treatments 36 blocks 12 8615 84 6. In particular, shows at pairwise non-isomorphic arrays OA(72,6,6,2).