Designs, Codes and Cryptography

, Volume 84, Issue 1–2, pp 261–281 | Cite as

Rank error-correcting pairs

Article
  • 174 Downloads

Abstract

Error-correcting pairs were introduced as a general method of decoding linear codes with respect to the Hamming metric using coordinatewise products of vectors, and are used for many well-known families of codes. In this paper, we define new types of vector products, extending the coordinatewise product, some of which preserve symbolic products of linearized polynomials after evaluation and some of which coincide with usual products of matrices. Then we define rank error-correcting pairs for codes that are linear over the extension field and for codes that are linear over the base field, and relate both types. Bounds on the minimum rank distance of codes and MRD conditions are given. Finally we show that some well-known families of rank-metric codes admit rank error-correcting pairs, and show that the given algorithm generalizes the classical algorithm using error-correcting pairs for the Hamming metric.

Keywords

Decoding Error-correcting pairs Linearized polynomials Rank metric Vector products 

Mathematics Subject Classification

15B33 94B35 94B65 

Notes

Acknowledgments

Rank error-correcting pairs of type II have been obtained in the case \( m = n \) independently by Alain Couvreur (About error correcting pairs. Personal communication, September 15, 2015). The authors wish to thank him for this communication and for useful discussions and comments. The authors wish to thank the anonymous reviewers for their helpful comments and also gratefully acknowledge the support from The Danish Council for Independent Research (Grant No. DFF-4002-00367). This paper was started during the visit of the second author to Aalborg University, Denmark, which was supported by the previous grant.

References

  1. 1.
    Berger T.P.: Isometries for rank distance and permutation group of Gabidulin codes. IEEE Trans. Inf. Theory 49(11), 3016–3019 (2003).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Boucher D., Geiselmann W., Ulmer F.: Skew-cyclic codes. Appl. Algebra Eng. Commun. Comput. 18(4), 379–389 (2007).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chaussade L., Loidreau P., Ulmer F.: Skew codes of prescribed distance or rank. Des. Codes Cryptogr. 50(3), 267–284 (2009).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25(3), 226–241 (1978).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Delsarte P.: On subfield subcodes of modified reed-solomon codes. IEEE Trans. Inf. Theory 21(5), 575–576 (2006).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Duursma I.M., Kötter R.: Error-locating pairs for cyclic codes. IEEE Trans. Inf. Theory 40(4), 1108–1121 (1994).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Duursma I.M., Pellikaan R.: A symmetric Roos bound for linear codes. J. Comb. Theory Ser. A 113(8), 1677–1688 (2006).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gabidulin E.M.: Theory of codes with maximum rank distance. Prob. Inf. Transm. 21, 3–16 (1985).MathSciNetMATHGoogle Scholar
  9. 9.
    Gabidulin E.M.: Rank q-cyclic and pseudo-q-cyclic codes. In: IEEE International Symposium on Information Theory (ISIT 2009), pp. 2799–2802 (2009).Google Scholar
  10. 10.
    Hartmann C.R.P., Tzeng K.K.: Generalizations of the BCH bound. Inf. Control 20(5), 489–498 (1972).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jurrius R.P.M.J., Pellikaan R.: The extended and generalized rank weight enumerator. ACA 2014, Applications of Computer Algebra, 9 July 2014, Fordham University, New York, Computer Algebra in Coding Theory and Cryptography (2014).Google Scholar
  12. 12.
    Jurrius R.P.M.J., Pellikaan R.: On defining generalized rank weights. arXiv:1506.02865 (2015).
  13. 13.
    Kötter R.: A unified description of an error locating procedure for linear codes. In: Proceedings of Algebraic and Combinatorial Coding Theory, Voneshta Voda, pp. 113–117 (1992).Google Scholar
  14. 14.
    Kshevetskiy A., Gabidulin E.M.: The new construction of rank codes. In: IEEE International Symposium on Information Theory (ISIT 2005), pp. 2105–2108 (2005).Google Scholar
  15. 15.
    Lidl R., Niederreiter H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20. Addison-Wesley, Amsterdam (1983).Google Scholar
  16. 16.
    Loidreau P.: A Welch–Berlekamp like algorithm for decoding Gabidulin codes. In: Ytrehus Ø. (ed.) Coding and Cryptography. Lecture Notes in Computer Science, vol. 3969, pp. 36–45. Springer, Berlin (2006).Google Scholar
  17. 17.
    Martínez-Peñas U.: On the roots and minimum rank distance of skew cyclic codes. Des. Codes Cryptogr. (2016). doi: 10.1007/s10623-016-0262-z.
  18. 18.
    Martínez-Peñas U.: On the similarities between generalized rank and Hamming weights and their applications to network coding. IEEE Trans. Inf. Theory 62(7), 4081–4095 (2016).MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Pellikaan R.: On decoding linear codes by error correcting pairs. Preprint, Eindhoven University of Technology (1988).Google Scholar
  20. 20.
    Pellikaan R.: On decoding by error location and dependent sets of error positions. Discret. Math. 106, 369–381 (1992).MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Pellikaan R.: On the existence of error-correcting pairs. J. Stat. Plan. Inference 51(2), 229–242 (1996). Shanghai Conference Issue on Designs, Codes, and Finite Geometries, Part I.Google Scholar
  22. 22.
    Ravagnani A.: Rank-metric codes and their duality theory. Des. Codes Cryptogr. 80(1), 197–216 (2015).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Roos C.: A generalization of the BCH bound for cyclic codes, including the Hartmann–Tzeng bound. J. Comb. Theory Ser. A 33, 229–232 (1982).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Roos C.: A new lower bound for the minimum distance of a cyclic code. IEEE Trans. Inf. Theory IT–29, 330–332 (1982).MathSciNetMATHGoogle Scholar
  25. 25.
    Silva D., Kschischang F.R.: On metrics for error correction in network coding. IEEE Trans. Inf. Theory 55(12), 5479–5490 (2009).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark
  2. 2.Department of Mathematics and Computing ScienceEindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations