Designs, Codes and Cryptography

, Volume 86, Issue 2, pp 303–318 | Cite as

Concatenation of convolutional codes and rank metric codes for multi-shot network coding

  • D. NappEmail author
  • R. Pinto
  • V. Sidorenko
Part of the following topical collections:
  1. Special Issue on Network Coding and Designs


In this paper we present a novel coding approach to deal with the transmission of information over a network. In particular we make use of the network several times (multi-shot) and impose correlation in the information symbols over time. We propose to encode the information via an inner and an outer code, namely, a Hamming metric convolutional code as an outer code and a rank metric code as an inner code. We show how this simple concatenation scheme can exploit the potential of both codes to produce a code that can correct a large number of error patterns.


Network coding Multi-shot network coding Concatenated codes Convolutional codes Rank metric codes 

Mathematics Subject Classification

94B05 94B20 94B10 15B33 11T71 



We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments. This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within Project UID/MAT/04106/2013.


  1. 1.
    Almeida P., Napp D., Pinto R.: A new class of superregular matrices and MDP convolutional codes. Linear Algebr. Appl. 439(7), 2145–2157 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Almeida P., Napp D., Pinto R.: Superregular matrices and applications to convolutional codes. Linear Algebr. Appl. 499, 1–25 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Badr A., Khisti A., Tan W.T., Apostolopoulos J.: Layered constructions for low-delay streaming codes. IEEE Trans. Inform. Theory (2013). arXiv:1308.3827v1.
  4. 4.
    Climent J., Napp D., Perea C., Pinto R.: Maximum distance separable 2D convolutional codes. IEEE Trans. Inform. Theory 62(2), 669–680 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Climent J., Napp D., Pinto R., Simões R.: Decoding of \(2{D}\) convolutional codes over the erasure channel. Adv. Math. Commun. 10(1), 179–193 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Combin. Theory Ser. A 25(3), 226–241 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gabidulin É.: Theory of codes with maximum rank distance. Probl. Inf. Transm. 21, 1–12 (1985).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gluesing-Luerssen H., Rosenthal J., Smarandache R.: Strongly MDS convolutional codes. IEEE Trans. Inform. Theory 52(2), 584–598 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guo W., Shi X., Cai N., Medard M.: Localized dimension growth: a convolutional random network coding approach to managing memory and decoding delay. IEEE Trans. Commun. 61(9), 3894–3905 (2013).CrossRefGoogle Scholar
  10. 10.
    Horlemann-Trautmann A., Marshall K.: New criteria for MRD and Gabidulin codes and some rank-metric code constructions. arXiv: 1507.08641.
  11. 11.
    Hutchinson R., Rosenthal J., Smarandache R., Trumpf J.: Convolutional codes with maximum distance profile. Syst. Control Lett. 54(1), 53–63 (2005).Google Scholar
  12. 12.
    Johannesson R., Zigangirov K.S.: Fundamentals of Convolutional Coding. IEEE Press, New York (1999).CrossRefzbMATHGoogle Scholar
  13. 13.
    Kötter R., Kschischang F.: Coding for errors and erasures in random network coding. IEEE Trans. Inform. Theory 54(8), 3579–3591 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    MacWilliams F.J., Sloane N.J.: The Theory of Error-Correcting Codes. North Holland, Amsterdam (1977).zbMATHGoogle Scholar
  15. 15.
    Mahmood R.: Rank metric convolutional codes with applications in network streaming. Master of Applied Science (2015).
  16. 16.
    Mahmood R., Badr A., Khisti A.: Streaming-codes for multicast over burst erasure channels. IEEE Trans. Inform. Theory 61(8), 4181–4208 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mahmood R., Badr A., Khisti A.: Convolutional codes with maximum column sum rank for network streaming. IEEE Trans. Inform. Theory to appear (2016).
  18. 18.
    McEliece R.J.: The algebraic theory of convolutional codes. In: Pless V., Huffman W. (eds.) Handbook of Coding Theory, vol. 1, pp. 1065–1138. Elsevier Science Publishers, Amsterdam (1998).Google Scholar
  19. 19.
    Napp D., Smarandache R.: Constructing strongly MDS convolutional codes with maximum distance profile. Adv. Math. Commun. 10(2), 275–290 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Napp D., Pinto R., Rosenthal J., Vettori P.: Rank metric convolutional codes. In: Proceedings of the 22nd International Symposium on Mathematical Theory of Network and Systems (MTNS), Minnesota, USA (2016).Google Scholar
  21. 21.
    Napp D., Pinto R., Toste T.: On MDS convolutional codes over \(\mathbb{Z}_{p^r}\). Des. Codes Cryptogr. pp. 1–14 (2016). doi: 10.1007/s10623-016-0204-9.
  22. 22.
    Nóbrega R., Uchoa-Filho B.: Multishot codes for network coding using rank-metric codes. In: IEEE Wireless Network Coding Conference (WiNC), pp. 1–6 (2010).Google Scholar
  23. 23.
    Pollara F., McEliece R.J., Abdel-Ghaffar K.: Finite-state codes. IEEE Trans. Inform. Theory 34(5), 1083–1089 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Prasad K., Rajan B.S.: Network error correction for unit-delay, memory-free networks using convolutional codes. IEEE Int. Conf. Commun. 2010, 1–6 (2010).Google Scholar
  25. 25.
    Rosenthal J.: Connections between linear systems and convolutional codes. In: Marcus B., Rosenthal J. (eds.) Codes, Systems and Graphical Models, IMA, vol. 123, pp. 39–66. Springer, New York (2001).CrossRefGoogle Scholar
  26. 26.
    Rosenthal J., Smarandache R.: Maximum distance separable convolutional codes. Appl. Algebr. Eng. Commun. Comput. 10(1), 15–32 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rosenthal J., York E.V.: BCH convolutional codes. IEEE Trans. Autom. Control 45(6), 1833–1844 (1999).MathSciNetzbMATHGoogle Scholar
  28. 28.
    Silva D., Kötter R., Kschischang F.: A rank-metric approach to error control in random network coding. IEEE Trans. Inform. Theory 54(9), 3951–3967 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tomas V., Rosenthal J., Smarandache R.: Decoding of convolutional codes over the erasure channel. IEEE Trans. Inform. Theory 58(1), 90–108 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wachter-Zeh A., Afanassiev V., Sidorenko V.R.: Fast decoding of Gabidulin codes. Des. Codes Cryptogr. 66, 57–73 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wachter-Zeh A., Stinner M., Sidorenko V.: Convolutional codes in rank metric with application to random network coding. IEEE Trans. Inform. Theory 61(6), 3199–3213 (2015).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.CIDMA - Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.Institute for Communications EngineeringTechnical University of MunichMunichGermany
  3. 3.Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

Personalised recommendations