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Theory of supports for linear codes endowed with the sum-rank metric

  • Umberto Martínez-PeñasEmail author
Article

Abstract

The sum-rank metric naturally extends both the Hamming and rank metrics in coding theory over fields. It measures the error-correcting capability of codes in multishot matrix-multiplicative channels (e.g. linear network coding or the discrete memoryless channel on fields). Although this metric has already shown to be of interest in several applications, not much is known about it. In this work, sum-rank supports for codewords and linear codes are introduced and studied, with emphasis on duality. The lattice structure of sum-rank supports is given; characterizations of the ambient spaces (support spaces) they define are obtained; the classical operations of restriction and shortening are extended to the sum-rank metric; and estimates (bounds and equalities) on the parameters of such restricted and shortened codes are found. Three main applications are given: (1) Generalized sum-rank weights are introduced, together with their basic properties and bounds; (2) It is shown that duals, shortened and restricted codes of maximum sum-rank distance (MSRD) codes are in turn MSRD; (3) Degenerateness and effective lengths of sum-rank codes are introduced and characterized. In an Appendix, skew supports are introduced, defined by skew polynomials, and their connection to sum-rank supports is given.

Keywords

Generalized sum-rank weights Hamming metric MSRD codes Multishot matrix-multiplicative channel Rank metric Sum-rank metric Sum-rank support Wire-tap channel 

Mathematics Subject Classification

94A60 94B05 94C99 

Notes

Acknowledgements

The author wishes to thank Frank R. Kschischang for valuable discussions on this manuscript. The author also whishes to thank the anonymous reviewers for the valuable comments on this work.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electrical & Computer EngineeringUniversity of TorontoTorontoCanada

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