Two $$n \times n$$ n × n Latin squares $$L_1, L_2$$ L 1 , L 2 are said to be orthogonal if, for every ordered pair ( x , y ) of symbols, there are coordinates ( i , j ) such that $$L_1(i,j) = x$$ L 1 ( i , j ) = x and $$L_2(i,j) = y$$ L 2 ( i , j ) = y . A k -MOLS is a sequence of k pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed k , log-asymptotically tight bounds on the number of k -MOLS. To study the situation when k grows with n , we bound the number of ways a k -MOLS can be extended to a $$(k+1)$$ ( k + 1 ) -MOLS. These bounds are again tight for constant k , and allow us to deduce upper bounds on the total number of k -MOLS for all k . These bounds are close to tight even for k linear in n , and readily generalise to the broader class of gerechte designs, which include Sudoku squares.

中文翻译：

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枚举相互正交拉丁方的扩展

两个 $$n \times n$$ n × n 拉丁方格 $$L_1, L_2$$ L 1 , L 2 被称为是正交的，如果对于每个有序的符号对 ( x , y )，有坐标 ( i , j ) 使得 $$L_1(i,j) = x$$ L 1 ( i , j ) = x 并且 $$L_2(i,j) = y$$ L 2 ( i , j ) = y 。一个k-MOLS是k个成对正交的拉丁方阵的序列，这些物体的存在和枚举引起了极大的关注。Keevash 和 Luria 的最新工作为所有固定的 k 提供了 k -MOLS 数量的对数渐近紧边界。为了研究 k 随 n 增长的情况，我们限制了 ak -MOLS 可以扩展到 $$(k+1)$$ ( k + 1 ) -MOLS 的方式。对于常数 k 来说，这些界限再次很紧，并允许我们推导出所有 k 的 k -MOLS 总数的上限。即使对于 n 中的 k 线性，这些边界也接近严格，