Designs, Codes and Cryptography

, Volume 83, Issue 3, pp 639–660 | Cite as

On the roots and minimum rank distance of skew cyclic codes

Article

Abstract

Skew cyclic codes play the same role as cyclic codes in the theory of error-correcting codes for the rank metric. In this paper, we give descriptions of these codes by root spaces, cyclotomic spaces and idempotent generators. We prove that the lattice of skew cyclic codes is anti-isomorphic to the lattice of root spaces, study these two lattices and extend the rank-BCH bound on their minimum rank distance to rank-metric versions of the van Lint–Wilson’s shift and Hartmann–Tzeng bounds. Finally, we study skew cyclic codes which are linear over the base field, proving that these codes include all Hamming-metric cyclic codes, giving then a new relation between these codes and rank-metric skew cyclic codes.

Keywords

Cyclic codes Finite rings Hamming distance Linearized polynomial rings Rank distance Skew cyclic codes 

Mathematics Subject Classification

15A03 15B33 94B15 

Notes

Acknowledgments

The author wishes to thank the anonymous reviewers for their very helpful comments and suggestions. The author also wishes to thank Olav Geil, Ruud Pellikaan and Diego Ruano for fruitful discussions and careful reading of the manuscript. Finally, the author gratefully acknowledges the support from The Danish Council for Independent Research (Grant No. DFF-4002-00367).

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark

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