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Designs, Codes and Cryptography

, Volume 82, Issue 1–2, pp 449–467 | Cite as

Algebraic decoding of folded Gabidulin codes

  • Hannes BartzEmail author
  • Vladimir Sidorenko
Article

Abstract

An efficient interpolation-based decoding algorithm for \(h\)-folded Gabidulin codes is presented that can correct rank errors beyond half the minimum rank distance for any code rate \(0\le R\le 1\). The algorithm serves as a list decoder or as a probabilistic unique decoder and improves upon existing schemes, especially for high code rates. A probabilistic unique decoder with adjustable decoding radius is presented. The decoder outputs a unique solution with high probability and requires at most \(\mathcal {O}({s^2n^2})\) operations in \(\mathbb {F}_{q^m}\), where \(1\le s\le h\) is a decoding parameter and \(n\le m\) is the length of the unfolded code over \(\mathbb {F}_{q^m}\). An upper bound on the average list size of folded Gabidulin codes and on the decoding failure probability of the decoder is given. Applying the ideas to a list decoding algorithm by Mahdavifar and Vardy (List-decoding of subspace codes and rank-metric codes up to Singleton bound, ISIT 2012) improves the performance when used as probabilistic unique decoder and gives an upper bound on the failure probability.

Keywords

Rank-metric codes Folded Gabidulin codes Probabilistic unique decoding Interpolation-based decoding 

Mathematics Subject Classification

Primary 94B35 94B05 

Notes

Acknowledgments

The authors would like to thank Gerhard Kramer, Joschi Brauchle and Johan S. R. Nielsen for fruitful discussions and helpful comments. H. Bartz was supported by the German Ministry of Education and Research in the framework of an Alexander von Humboldt-Professorship.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute for Communications EngineeringTechnical University of MunichMunichGermany
  2. 2.Institute for Information Transmission ProblemsRussian Academy of ScienceMoscowRussia

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