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Designs, Codes and Cryptography

, Volume 82, Issue 1–2, pp 389–409 | Cite as

Row reduction applied to decoding of rank-metric and subspace codes

  • Sven Puchinger
  • Johan Rosenkilde né Nielsen
  • Wenhui Li
  • Vladimir Sidorenko
Article
  • 221 Downloads

Abstract

We show that decoding of \(\ell \)-Interleaved Gabidulin codes, as well as list-\(\ell \) decoding of Mahdavifar–Vardy (MV) codes can be performed by row reducing skew polynomial matrices. Inspired by row reduction of \(\mathbb {F}[x]\) matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into a certain reduced form. We apply this to solve generalised shift register problems over skew polynomial rings which occur in decoding \(\ell \)-Interleaved Gabidulin codes. We obtain an algorithm with complexity \(O(\ell \mu ^2)\) where \(\mu \) measures the size of the input problem and is proportional to the code length n in the case of decoding. Further, we show how to perform the interpolation step of list-\(\ell \)-decoding MV codes in complexity \(O(\ell n^2)\), where n is the number of interpolation constraints.

Keywords

Skew polynomials Row reduction Module minimisation Gabidulin codes Shift register synthesis Mahdavifar–Vardy codes 

Mathematics Subject Classification

12Y05 12E15 11T71 

Notes

Acknowledgments

The authors would like to thank the anonymous reviewers for suggestions that have substantially improved the clarity of the paper. Sven Puchinger (Grant BO 867/29-3), Wenhui Li and Vladimir Sidorenko (Grant BO 867/34-1) were supported by the German Research Foundation “Deutsche Forschungsgemeinschaft” (DFG).

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Sven Puchinger
    • 1
  • Johan Rosenkilde né Nielsen
    • 2
  • Wenhui Li
    • 1
  • Vladimir Sidorenko
    • 3
  1. 1.Institute of Communications EngineeringUlm UniversityUlmGermany
  2. 2.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKongens LyngbyDenmark
  3. 3.Institute for Communications EngineeringTU MünchenMunichGermany

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