In this paper, we develop a geometric framework for matrix rank-metric codes based on generator tensors and their slice spaces. To every nondegenerate matrix rank-metric code, we associate two systems, which translate metric properties of the code into geometric conditions involving intersections with hyperplanes. This leads to a correspondence between equivalence classes of nondegenerate matrix rank-metric codes and equivalence classes of systems, as well as to Delsarte-type incidence identities relating the rank distribution of a code over a finite field to those of its associated systems. As an application, we introduce generalized weights through the notion of evasive systems, study faithful and one-weight codes over finite fields, and recover known bounds and results from the theory of semifields. Finally, we use this framework to associate additive Hamming-metric codes with matrix rank-metric codes and show that several metric properties are preserved under this correspondence.

翻译：本文基于生成张量及其切片空间，为矩阵秩度量码建立了一个几何框架。针对每个非退化矩阵秩度量码，我们关联两个系统，将码的度量性质转化为涉及超平面交集的几何条件。这推导出非退化矩阵秩度量码的等价类与系统等价类之间的对应关系，以及将有限域上码的秩分布与其关联系统的秩分布联系起来的Delsarte型关联恒等式。作为应用，我们通过规避系统的概念引入广义权重，研究有限域上的忠实码和单权重码，并恢复半域理论中的已知界与结论。最后，我们利用该框架将加法海明度量码与矩阵秩度量码关联起来，并证明在此对应关系下若干度量性质保持不变。