A new rank metric for convolutional codes
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A new rank metric for convolutional codes P. Almeida1 · D. Napp2 Received: 27 February 2020 / Revised: 22 June 2020 / Accepted: 25 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let F[D] be the polynomial ring with entries in a finite field F. Convolutional codes are submodules 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 new metric 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. Keywords Convolutional codes · Rank metric · Column distance · Network coding · Maximum distance profile Mathematics Subject Classification 15B33 · 15B05 · 94B10
1 Introduction Within the area of coding theory, network coding has been a very active topic of research as it provides an effective tool to disseminate information (packets) over networks. Mathe-
Communicated by M. Lavrauw.
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D. Napp [email protected] P. Almeida [email protected]
1
CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal
2
Department of Mathematics, University of Alicante, Alicante, Spain
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P. Almeida, D. Napp
matically, we can consider the transmitted packets as columns of a matrix with entries in a finite field Fq , and the linear combinations performed in the nodes of the network correspond to columns operations on this matrix. If no errors occur during the transmission over such a network, the Fq -column space of the transmitted matrix remains invariant. To achieve a reliable communication over this channel, matrix codes are employed forming the so-called rank metric codes (see [31]). Rank metric codes such as Gabidulin codes are known to be able to protect packets in such a scenario. We call these codes one-shot codes, as they use the (network) channel only once. However, coding can also be performed over multiple uses of the network (multi-shot) in a sequential fashion, as it has been recently shown by several authors, see for instance [5,17,22,26,32]. The general idea of multi-shot network coding stems from the f
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