We show that any partial Latin square of order $n$ can be embedded in a Latin square of order at most $16n^2$ which has at least $2n$ mutually orthogonal mates. We also show that for any $t\geq 2$, a pair of orthogonal partial Latin squares of order $n$ can be embedded into a set of $t$ mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to $n$. Furthermore, the constructions that we provide show that MOLS($n^2$)$\geq$MOLS($n$)+2, consequently we give a set of $9$ MOLS($576$). The maximum known size of a set of MOLS($576$) was previously given as $8$ in the literature.

中文翻译：

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在具有许多相互正交的配对的拉丁方中嵌入部分拉丁方

我们表明，任何 $n$ 阶的部分拉丁方都可以嵌入到最多 $16n^2$ 的拉丁方中，该拉丁方至少具有 $2n$ 相互正交的配对。我们还表明，对于任何 $t\geq 2$，一对 $n$ 阶的正交偏拉丁方可以嵌入到一组 $t$ 相互正交的拉丁方（MOLS）中，该集合的阶多项式关于 $ n$。此外，我们提供的构造表明 MOLS($n^2$)$\geq$MOLS($n$)+2，因此我们给出了一组 $9$ MOLS($576$)。一组 MOLS 的最大已知大小（$576$）以前在文献中给出为 $8$。