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. It is known that a uniform $(n\times n)/k$ semi-Latin square is Schur optimal in the class of all $(n\times n)/k$ semi-Latin squares, and here 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 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 dual of) 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 $n-1$ 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 $(6\times 6)/10$ 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.

中文翻译：

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均匀半拉丁方及其成对方差像差

对于整数 $n>2$ 和 $k>0$，$(n\times n)/k$ 半拉丁方是一个 $n\times n$ 数组的 $k$-子集（称为块）的$nk$-set（处理），这样每个处理在数组的每一行和每列中出现一次。如果每对块（不在同一行或列中）在相同的正数处理中相交，则半拉丁方是统一的。已知一个统一的 $(n\times n)/k$ 半拉丁方阵在所有的 $(n\times n)/k$ 半拉丁方阵中是 Schur 最优的，这里我们证明当 a均匀$(n\times n)/k$半拉丁方阵存在，Schur最优$(n\times n)/k$半拉丁方阵正是一致的。然后，我们使用 JP Morgan 为仿射可解析设计引入的成对方差 (PV) 像差标准来比较均匀半拉丁方，当存在$n-1$相互正交的$n$阶拉丁方阵时，求出具有最小PV像差的统一$(n\times n)/k$半拉丁方阵。当 $n=6$ 时这些不存在，并且在这种情况下最小的均匀半拉丁方格的大小为 $(6\times 6)/10$。我们提供了统一的 $(6\times 6)/10$ 半拉丁方格的完整分类，并显示了具有最小 PV 像差的（的对偶）。当存在 $n-1$ 个相互正交的拉丁方阶时，我们给出一个构造，产生一个统一的 $((n+1)\times (n+1))/((n-2)n)$ 半拉丁方$n$，并确定这种均匀半拉丁方阵的 PV 像差。最后，我们描述了某些仿射可解析设计和平衡的不完全块设计如何从统一的半拉丁方阵构造出来，