This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce q-polymatroids, the q-analogue of polymatroids, and develop their basic properties. We associate a pair of q-polymatroids with a rank-metric code and show that several invariants and structural properties of the code, such as generalized weights, the property of being MRD or an optimal anticode, and duality, are captured by the associated combinatorial object.

中文翻译：

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秩度量代码和q-多类拟阵

本文从代数和组合的角度为秩度量代码的研究做出了贡献。我们介绍了q -polymatroids，即q- polymatroids的类似物，并开发了它们的基本特性。我们将一对q -polymatroids与等级度量代码相关联，并显示该代码的几个不变性和结构特性，例如广义权重，作为MRD或最佳反码的特性以及对偶性，都由相关的组合捕获。目的。