A lower bound is presented for the minimal number of filled cells in a maximal partial Latin hypercube of dimension d and order n. The result generalises and extends previous results for \(d=2\) (Latin squares) and \(d=3\) (Latin cubes). Explicit constructions show that this bound is near-optimal for large \(n> d\). For \(d>n\), a connection with Hamming codes shows that this lower bound gives a related upper bound for the same quantity. The results can be interpreted in terms of independent dominating sets in certain graphs, and in terms of codes that have covering radius 1 and minimum distance at least 2.

中文翻译：

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关于最大部分拉丁超立方体

给出了维度d和阶数n的最大部分拉丁超立方体中填充单元的最小数量的下界。该结果概括并扩展了先前的\(d=2\)（拉丁方）和\(d=3\)（拉丁方）结果。显式构造表明，对于较大的\(n> d\) ，此界限接近最佳。对于\(d>n\)，与汉明码的连接表明该下界给出了相同数量的相关上限。结果可以根据某些图中的独立支配集以及覆盖半径为 1 且最小距离至少为 2 的代码来解释。