In this paper, a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative principal ideal ring to be a metric is obtained. We completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring where the homogeneous distance is a metric. Furthermore, the minimum homogeneous distances of the duals of such codes are also explicitly investigated.

中文翻译：

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有限交换主理想环上的齐次度量和矩阵乘积码

在本文中，获得了将任意有限可交换主理想环上的均匀距离作为度量的充要条件。我们完全刻画了在任意有限主理想环上矩阵乘积码的齐次距离的下界，其中齐次距离是度量。此外，还明确研究了这种代码的对偶的最小均匀距离。