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

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Abstract

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.

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References

  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).

    Article  MathSciNet  MATH  Google Scholar 

  2. Almeida P., Napp D., Pinto R.: Superregular matrices and applications to convolutional codes. Linear Algebr. Appl. 499, 1–25 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  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. Climent J., Napp D., Perea C., Pinto R.: Maximum distance separable 2D convolutional codes. IEEE Trans. Inform. Theory 62(2), 669–680 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  6. Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Combin. Theory Ser. A 25(3), 226–241 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  7. Gabidulin É.: Theory of codes with maximum rank distance. Probl. Inf. Transm. 21, 1–12 (1985).

    MathSciNet  MATH  Google Scholar 

  8. Gluesing-Luerssen H., Rosenthal J., Smarandache R.: Strongly MDS convolutional codes. IEEE Trans. Inform. Theory 52(2), 584–598 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  Google Scholar 

  10. Horlemann-Trautmann A., Marshall K.: New criteria for MRD and Gabidulin codes and some rank-metric code constructions. arXiv: 1507.08641.

  11. Hutchinson R., Rosenthal J., Smarandache R., Trumpf J.: Convolutional codes with maximum distance profile. Syst. Control Lett. 54(1), 53–63 (2005).

  12. Johannesson R., Zigangirov K.S.: Fundamentals of Convolutional Coding. IEEE Press, New York (1999).

    Book  MATH  Google Scholar 

  13. Kötter R., Kschischang F.: Coding for errors and erasures in random network coding. IEEE Trans. Inform. Theory 54(8), 3579–3591 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  14. MacWilliams F.J., Sloane N.J.: The Theory of Error-Correcting Codes. North Holland, Amsterdam (1977).

    MATH  Google Scholar 

  15. Mahmood R.: Rank metric convolutional codes with applications in network streaming. Master of Applied Science (2015). https://tspace.library.utoronto.ca/handle/1807/70480.

  16. Mahmood R., Badr A., Khisti A.: Streaming-codes for multicast over burst erasure channels. IEEE Trans. Inform. Theory 61(8), 4181–4208 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  17. Mahmood R., Badr A., Khisti A.: Convolutional codes with maximum column sum rank for network streaming. IEEE Trans. Inform. Theory to appear (2016). http://arxiv.org/pdf/1506.03792.pdf.

  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. Napp D., Smarandache R.: Constructing strongly MDS convolutional codes with maximum distance profile. Adv. Math. Commun. 10(2), 275–290 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  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).

  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. 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).

  23. Pollara F., McEliece R.J., Abdel-Ghaffar K.: Finite-state codes. IEEE Trans. Inform. Theory 34(5), 1083–1089 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  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. 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).

    Chapter  Google Scholar 

  26. Rosenthal J., Smarandache R.: Maximum distance separable convolutional codes. Appl. Algebr. Eng. Commun. Comput. 10(1), 15–32 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  27. Rosenthal J., York E.V.: BCH convolutional codes. IEEE Trans. Autom. Control 45(6), 1833–1844 (1999).

    MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  29. Tomas V., Rosenthal J., Smarandache R.: Decoding of convolutional codes over the erasure channel. IEEE Trans. Inform. Theory 58(1), 90–108 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  30. Wachter-Zeh A., Afanassiev V., Sidorenko V.R.: Fast decoding of Gabidulin codes. Des. Codes Cryptogr. 66, 57–73 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal

    D. Napp & R. Pinto

  2. Institute for Communications Engineering, Technical University of Munich, Munich, Germany

    V. Sidorenko

  3. Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

    V. Sidorenko

Authors
  1. D. Napp
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  2. R. Pinto
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  3. V. Sidorenko
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Correspondence to D. Napp.

Additional information

This is one of several papers published in Designs, Codes and Cryptography comprising the Special Issue on Network Coding and Designs.

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Napp, D., Pinto, R. & Sidorenko, V. Concatenation of convolutional codes and rank metric codes for multi-shot network coding. Des. Codes Cryptogr. 86, 303–318 (2018). https://doi.org/10.1007/s10623-017-0346-4

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  • DOI: https://doi.org/10.1007/s10623-017-0346-4

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