In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors.

中文翻译：

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系统最大和秩代码

在过去的十年中，人们对将配备汉明度量的代码的结果扩展为带有秩度量的代码的类似结果产生了极大的兴趣。这项工作遵循这一研究思路，研究具有最大秩距离的代码的系统生成矩阵（编码器）的表征。在汉明距离的背景下，这些代码是所谓的最大距离可分 (MDS) 代码，并且系统编码器已得到充分研究。在本文中，我们以最大秩距离 (MRD) 码和最大和秩距离 (MSRD) 码的系统形式研究编码器的代数特性和表示。我们分别处理分组码和卷积码，并以系统形式为编码器提供必要和充分条件，以生成具有最大（和）秩距离的代码。这些特征是根据某些矩阵给出的，这些矩阵在扩展域中必须是超正则的，并且在基域上执行一些变换后保持超正则性。我们总结了在小场上展示最大和秩卷积码的一些例子的工作。对于给定的参数，获得的示例比其他作者获得的示例更小。我们总结了在小场上展示最大和秩卷积码的一些例子的工作。对于给定的参数，获得的例子比其他作者获得的例子更小。我们总结了在小场上展示最大和秩卷积码的一些例子的工作。对于给定的参数，获得的示例比其他作者获得的示例更小。