MOLS完备集的正交交换
Orthogonal trades in complete sets of MOLS
摘要：
设$B_p$是由整数模奇素数$p$的加法表给出的拉丁方。在这里，我们考虑$B_p$中拉丁交易的属性，这些属性与有限域构造给出的其中一个$p-1$ MOLS保持正交性。我们表明，对于某些选择的正交伴侣，有一个下限对数$p$的次数，每个符号出现在这样的贸易，与整体下限的$(\log{p})^2/\log\log{p}$的大小，这样的贸易。这种交换意味着循环群的正形的存在，它与线性正形有很小的不同。我们还表明，在$B_p$中的任何横向击中主对角线要么$p$或最多$p-\log_2{p}-1$倍。最后，如果$p\equiv 1\mod{6}$，我们证明了拉丁方的存在性，它包含一个与$B_p$正交的$2\times 2$子方。
英文摘要：
Let $B_p$ be the Latin square given by the addition table for the integers modulo an odd prime $p$. Here we consider the properties of Latin trades in $B_p$ which preserve orthogonality with one of the $p-1$ MOLS given by the finite field construction. We show that for certain choices of the orthogonal mate, there is a lower bound logarithmic in $p$ for the number of times each symbol occurs in such a trade, with an overall lower bound of $(\log{p})^2/\log\log{p}$ for the size of such a trade. Such trades imply the existence of orthomorphisms of the cyclic group which differ from a linear orthomorphism by a small amount. We also show that any transversal in $B_p$ hits the main diagonal either $p$ or at most $p-\log_2{p}-1$ times. Finally, if $p\equiv 1\mod{6}$ we show the existence of Latin square containing a $2\times 2$ subsquare which is orthogonal to $B_p$.
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