论文标题
系统的最大总和排名代码
Systematic Maximum Sum Rank Codes
论文作者
论文摘要
在过去的十年中，人们对扩展配备有锤子指标的代码的结果非常感兴趣，以对具有等级度量的代码进行类似的结果。这项工作遵循了这一研究线索，并研究了最大等级距离的代码的系统发电机矩阵（编码器）的表征。在锤距离的情况下，这些代码是所谓的最大距离可分离（MDS）代码，并且已经完全研究了系统的编码器。在本文中，我们研究了最大秩距离（MRD）代码的系统形式的代数属性和表示编码器的表示。我们分别解决块代码和卷积代码，并为系统形式的编码器提供必要的和足够的条件，以生成具有最大（SUM）等级距离的代码。这些特征是根据某些矩阵给出的，这些矩阵在扩展场中必须是超规则的，并且在基础场上进行了一些转换后保持超规范性。我们总结了一些示例的工作，其中一些示例是小领域上最高总和卷积代码的示例。对于给定参数，所获得的示例比其他作者获得的示例较小。
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.
