论文标题
均匀的半拉丁正方形及其成对变异畸变
Uniform semi-Latin squares and their pairwise-variance aberrations
论文作者
论文摘要
对于整数，$ n> 2 $和$ k> 0 $，a $（n \ times n）/k $半拉丁蛋白方形是$ n \ times n $阵列的$ k $ -subsets（称为块）（处理的块）（治疗）（治疗），因此每个处理量均为每一排行，每列在每一列中都有一次。如果每对不在同一行或列中的每对块中的每对块在相同的阳性处理中相交，则半拉丁蛋白正方形是均匀的。我们表明，当存在一个均匀的$（n \ times n）/k $半拉丁平方时，Schur Optimal $（n \ times n）/k $ semi-latin正方形正是统一的。 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 $n$.当$ n = 6 $时，这些不存在，在这种情况下，最小的统一半拉蛋白正方形具有尺寸$（6 \ times 6）/10 $。我们提出了均匀$（6 \ times 6）/10 $半拉丁蛋白正方形的完整分类，并显示一个最少的PV像差。当存在$ n-1 $ n $ n $ n $ n $ n $ n $ n $ n $ n $时，我们给出了产生统一$（（n+1）\ times（n+1））的统一$（（n+1）\ times（n+1））的结构，并确定这种均匀的半拉丁蛋白平方的PV像差。最后，我们描述了如何通过均匀的半拉蛋白正方形构建某些可分解的设计和平衡不完全块设计（BIBDS）。从我们分类的均匀$（6 \ times 6）/10 $半平移平方中，我们获得（最高块设计同构），正好是16875 16875可分解的仿生设计，用于在36个尺寸12和8615块的36个块中的72次处理，用于36个尺寸和84个处理的bibds bibds，尤其是6号尺寸的36个块。 $ \ mathrm {oa}（72,6,6,2）$。
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 $\mathrm{OA}(72,6,6,2)$.
