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A new rank metric for convolutional codes

机译：卷积码的新级别度量

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摘要

Let F[D] be the polynomial ring with entries in a finite field F. Convolutional codes are sub-modules of F[D](n) that can be described by left prime polynomial matrices. In the last decade there has been a great interest in convolutional codes equipped with a rank metric, called sum rank metric, due to their wide range of applications in reliable linear network coding. However, this metric suits only for delay free networks. In this work we continue this thread of research and introduce a newmetric that overcomes this restriction and therefore is suitable to handle more general networks. We study this metric and provide characterizations of the distance properties in terms of the polynomial matrix representations of the convolutional code. Convolutional codes that are optimal with respect to this new metric are investigated and concrete constructions are presented. These codes are the analogs of Maximum Distance Profile convolutional codes in the context of network coding. Moreover, we show that they can be built upon a class of superregular matrices, with entries in an extension field, that preserve their superregularity properties even after multiplication with some matrices with entries in the ground field.
机译：让F [D]具有有限磁场F的条目的多项式环。卷积码是F [D]（n）的子模块，其可以由左主要多项式矩阵描述。在过去的十年中，由于它们在可靠的线性网络编码中的广泛应用范围内，对配备的卷积指数有很大的兴趣，称为SUM秩规范。但是，此度量仅适用于延迟免费网络。在这项工作中，我们继续这项研究线程，并介绍了一种克服这种限制，因此适合处理更多通用网络。我们研究了该度量，并在卷积码的多项式矩阵表示方面提供距离特性的特征。研究了关于该新度量最佳的卷积码，并提出了混凝土结构。这些代码是网络编码的上下文中的最大距离轮廓卷积码的类似物。此外，我们表明它们可以建立在一类超级矩阵，其中延伸字段中的条目，即使在乘以带有地面中的条目中的一些矩阵之后也保持其超级性能。

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