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On Error Control Codes for Random Network Coding

  • Rudolf Ahlswede
Chapter
Part of the Foundations in Signal Processing, Communications and Networking book series (SIGNAL, volume 15)

Abstract

The random network coding approach is approved to be an effective technique for linear network coding, however is highly susceptible to errors and adversarial attacks. Recently Kötter and Kschischang [1] introduced the operator channel, where the inputs and outputs are subspaces of a given vector space, showing that this is a natural transmission model in noncoherent random network coding. A suitable metric, defined for subspaces: \(d(U,V)=\dim U+\dim V-2\dim (U\cap V)\), gives rise to the notion of codes capable of correcting (different kinds of) errors in noncoherent random network coding. In this lecture we continue the study of coding for operator channels started in [1]. Bounds and constructions for codes correcting insertions/deletions are presented.

References

  1. 1.
    R. Kötter, F.R. Kschischang, Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory 54(8), 3579–3591 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    T. Ho, R. Kötter, M. Medard, D.R. Karger, M. Effros, The benefits of coding over routing in randomized setting, in Proceedings of the 2003 IEEE International Symposium on Information Theory, (Yokohama), June 20–July 3 (2003), p. 442Google Scholar
  3. 3.
    P.A. Chou, Y. Wu, K. Jain, Practical network coding, in Proceedings of the 2003 Allerton Conference on Communication Control and Computing, (Monticello, IL) (2003)Google Scholar
  4. 4.
    T. Ho, M. Medard, R. Kötter, D.R. Karger, M. Effros, J. Shi, B. Leong, A random linear network coding approach to multicast. IEEE Trans. Inf. Theory 52(10), 4413–4430 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    N. Cai, R.W. Yeung, Network error correction, part II: lower bounds. Commun. Inf. Syst. 6(1), 37–54 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    R.W. Yeung, N. Cai, Network error correction, part I: basic concepts and upper bounds. Commun. Inf. Syst. 6(1), 19–36 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance Regular Graphs (Springer, Berlin, 1989)CrossRefGoogle Scholar
  8. 8.
    P. Delsarte, An algebraic approach to association schemes of coding theory. Philips J. Res. 10 (1973)Google Scholar
  9. 9.
    R. Ahlswede, H. Aydinian, L.H. Khachatrian, On perfect codes and related concepts. Des. Codes Cryptogr. 22(3), 221–237 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    E.M. Gabidulin, Theory of codes with maximum rank distance. Probl. Inf. Transm. 21(1), 1–12 (1985)MathSciNetzbMATHGoogle Scholar
  11. 11.
    H. Wang, C. Xing, R. Safavi-Naimi, Linear authentification codes: bounds and constructions. IEEE Trans. Inf. Theory 49, 866–872 (2003)CrossRefGoogle Scholar
  12. 12.
    T. Etzion, A. Vardy, Error-correcting codes in projective spaces, in Proceedings of the IEEE International Symposium on Information Theory, Toronto (2008), pp. 871–875Google Scholar
  13. 13.
    P. Frankl, R.M. Wilson, The Erdös-Ko-Rado theorem for vector spaces. J. Comb. Theory Ser. A 43, 228–236 (1986)CrossRefGoogle Scholar
  14. 14.
    M. Schwartz, T. Etzion, Codes and anticodes in the Grassman graph. J. Comb. Theory Ser. A 97(1), 27–42 (2002)MathSciNetCrossRefGoogle Scholar
  15. 15.
    S.-T. Xia, F.-W. Fu, Johnson type bounds on constant dimension codes. Des. Codes Cryptogr. 50, 163–172 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    F.J. MacWilliams, N.J.A. Sloane, The Theory of Error Correcting Codes (North-Holland, Amsterdam, 1977)zbMATHGoogle Scholar
  17. 17.
    D.J. Kleitman, On an extremal property of antichains in partial orders. The LYM property and some of its implications and applications, in Combinatorics, Part 2: Graph Theory; Foundations, Partitions and Combinatorial Geometry. Mathematical Centre tracts, vol. 56 (Mathematisch Centrum, Amsterdam, 1974), pp. 77–90Google Scholar
  18. 18.
    D. Silva, F.R. Kschischang, R. Koetter, A rank-metric approach to error control in random network coding. IEEE Trans. Inf. Theory 54(9), 3951–3967 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    T. Etzion, N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagram. CoRR (2008), arXiv:0807.4846

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.BielefeldGermany

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