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Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 547–567 | Cite as

Improved syndrome decoding of lifted \(L\)-interleaved Gabidulin codes

  • Hannes BartzEmail author
  • Vladimir Sidorenko
Article
  • 86 Downloads
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography

Abstract

A syndrome decoding algorithm for lifted interleaved Gabidulin codes of order L is proposed. The algorithm corrects L times more deviations (packet insertions) than known syndrome decoding methods with probability at least \(1-8q^{-n}\), where n is the length of the (interleaved) Gabidulin code. For \(n<L\), the proposed scheme has L times less computational complexity than known interpolation-factorization based decoders which attain the same decoding region. Upper bounds on the decoding failure probability are derived. Up to our knowledge this is the first syndrome-based scheme for interleaved subspace codes that can correct deviations beyond the unique decoding radius.

Keywords

Subspace codes Rank-metric codes Interleaved Gabidulin codes Probabilistic unique decoding Syndrome-based decoding 

Mathematics Subject Classification

Primary 94B35 94B05 

Notes

Acknowledgements

The authors would like to thank Manuela Meier for developing the simulation framework for the improved syndrome decoder.

References

  1. 1.
    Ahlswede R., Cai N., Li S., Yeung R.: Network information flow. IEEE Trans. Inf. Theory 46(4), 1204–1216 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bartz H.: Algebraic decoding of subspace and rank-metric codes. PhD thesis, Technische Universität München (2017)Google Scholar
  3. 3.
    Bartz H., Meier M., Sidorenko V.: Improved syndrome decoding of interleaved subspace codes. In: 11th International ITG Conference on Systems, Communication and Coding 2017 (SCC), Hamburg, Germany (2017).Google Scholar
  4. 4.
    Bartz H., Wachter-Zeh A.: Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases. Adv Math Commun. 12(4), 773–804 (2018).  https://doi.org/10.3934/amc.2018046.CrossRefzbMATHGoogle Scholar
  5. 5.
    Gabidulin E.M.: Theory of codes with maximum rank distance. Probl. Inf. Transm. 21(1), 3–16 (1985).MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gabidulin E.M., Paramonov A.V., Tretjakov O.V.: Rank errors and rank erasures correction. In: International Colloquium Coding Theory (1991).Google Scholar
  7. 7.
    Gabidulin E.M., Pilipchuk N.I.: A new method of erasure correction by rank codes. In: IEEE International Symposium of Information Theory (ISIT), p. 423 (2003).Google Scholar
  8. 8.
    Gabidulin E.M., Pilipchuk N.I.: Error and erasure correcting algorithms for rank codes. Des. Codes Cryptogr. 49(1–3), 105–122 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gadouleau M., Yan Z.: Complexity of decoding Gabidulin codes. In: 42nd Annual Conference on Information Sciences and Systems (CISS), pp. 1081–1085 (2008).  https://doi.org/10.1109/CISS.2008.4558679.
  10. 10.
    Guruswami V., Xing C.: List decoding Reed-Solomon, algebraic-geometric, and Gabidulin subcodes up to the singleton bound. Electron. Colloq. Comput. Complex. 19, 146 (2012).zbMATHGoogle Scholar
  11. 11.
    Horn R.A., Johnson C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012).CrossRefGoogle Scholar
  12. 12.
    Kötter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory 54(8), 3579–3591 (2008).  https://doi.org/10.1109/TIT.2008.926449.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Li W., Sidorenko V., Silva D.: On transform-domain error and erasure correction by Gabidulin codes. Des. Codes Cryptogr. 73, 571–586 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lidl R., Niederreiter H.: Finite Fields. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1996).zbMATHGoogle Scholar
  15. 15.
    Loidreau P., Overbeck R.: Decoding rank errors beyond the error correcting capability. In: International Workshop Algebraic and Combinatorial Coding Theory (ACCT), pp. 186–190 (2006).Google Scholar
  16. 16.
    Overbeck R.: Decoding interleaved Gabidulin codes and ciphertext-security for GPT variants (preprint).Google Scholar
  17. 17.
    Overbeck R.: Public key cryptography based on coding theory. PhD thesis, TU Darmstadt, Darmstadt, Germany (2007).Google Scholar
  18. 18.
    Richter G., Plass S.: Error and erasure decoding of rank-codes with a modified Berlekamp-Massey algorithm. In: ITG Conference on Source Channel Coding (SCC) (2004).Google Scholar
  19. 19.
    Sidorenko V.R., Jiang L., Bossert M.: Skew-feedback shift-register synthesis and decoding interleaved Gabidulin codes. IEEE Trans. Inf. Theory 57(2), 621–632 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sidorenko V.R., Richter G., Bossert M.: Linearized shift-register synthesis. IEEE Trans. Inf. Theory 57(9), 6025–6032 (2011).  https://doi.org/10.1109/TIT.2011.2162173.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Silva D.: Error control for network coding. PhD thesis, University of Toronto, Toronto, Canada (2009).Google Scholar
  22. 22.
    Silva D., Kschischang F.R., Kötter R.: A rank-metric approach to error control in random network coding. IEEE Trans. Inf. Theory 54(9), 3951–3967 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Skachek V., Roth R.M.: Probabilistic algorithm for finding roots of linearized polynomials. Des. Codes Cryptogr. 46(1), 17–23 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wachter-Zeh A., Zeh A.: Interpolation-based decoding of interleaved Gabidulin codes. In: International Workshop on Coding and Cryptography (WCC) (2013).Google Scholar
  25. 25.
    Wachter-Zeh A., Zeh A.: List and unique error-erasure decoding of interleaved Gabidulin codes with interpolation techniques. Des. Codes Cryptogr. 73(2), 547–570 (2014).  https://doi.org/10.1007/s10623-014-9953-5.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Communications and NavigationGerman Aerospace Center (DLR)OberpfaffenhofenGermany
  2. 2.Institute for Communications EngineeringTechnical University of Munich (TUM)MunichGermany

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