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A new rank metric for convolutional codes

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Abstract

Let \({\mathbb {F}}[D]\) be the polynomial ring with entries in a finite field \({\mathbb {F}}\). Convolutional codes are submodules of \({\mathbb {F}}[D]^n\) 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.

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Acknowledgements

The first author was supported by The Center for Research and Development in Mathematics and Applications (CIDMA), through the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020. The second author work was supported by the Ministerio de Ciencia e Innovación, Gobierno de España, under the grant PID2019-108668GB-I00.

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Authors and Affiliations

  1. CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal

    P. Almeida

  2. Department of Mathematics, University of Alicante, Alicante, Spain

    D. Napp

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  1. P. Almeida
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  2. D. Napp
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Correspondence to D. Napp.

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Communicated by M. Lavrauw.

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Almeida, P., Napp, D. A new rank metric for convolutional codes. Des. Codes Cryptogr. 89, 53–73 (2021). https://doi.org/10.1007/s10623-020-00808-w

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