Wuhan University Journal of Natural Sciences

, Volume 20, Issue 5, pp 391–396 | Cite as

Exact decoding probability of random linear network coding for combinatorial networks

Computer Science


Combinatorial networks are widely applied in many practical scenarios. In this paper, we compute the closed-form probability expressions of successful decoding at a sink and at all sinks in the multicast scenario, in which one source sends messages to k destinations through m relays using random linear network coding over a Galois field. The formulation at a (all) sink(s) represents the impact of major parameters, i.e., the size of field, the number of relays (and sinks) and provides theoretical groundings to numerical results in the literature. Such condition maps to the receivers’ capability to decode the original information and its mathematical characterization is helpful to design the coding. In addition, numerical results show that, under a fixed exact decoding probability, the required field size can be minimized.


random linear network coding successful probability combinatorial networks 

CLC number

TP 391.4 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Yeung R W, Zhang Z. Distributed source coding for satellite communications [J]. IEEE Transactions on Information Theory, 1999, 45(4): 1111–1120.CrossRefGoogle Scholar
  2. [2]
    Ahlswede R, Cai N, Li S Y, et al. Network information flow[J]. IEEE Transactions on Information Theory, 2000, 46(4): 1204–1216.CrossRefGoogle Scholar
  3. [3]
    Li S Y, Yeung R W, Cai N. Linear network coding[J]. IEEE Transactions on Information Theory, 2003, 49(2): 371–381.CrossRefGoogle Scholar
  4. [4]
    Koetter R, Médard M. An algebraic approach to network coding[J]. IEEE/ACM Trans Network, 2003,11(5): 782–795.CrossRefGoogle Scholar
  5. [5]
    Jaggi S, Sanders P, Chou P A, et al. Polynomial time algorithms for multicast network code construction[J]. IEEE Transactions on Information Theory, 2005, 51(6): 1973–1982.CrossRefGoogle Scholar
  6. [6]
    Fragouli C, Soljanin E. Network Coding Fundamentals [M]. Boston-Delft: Now Publishers Inc, 2007.Google Scholar
  7. [7]
    Yeung R W. Information Theory and Network Coding[M]. New York: Springer-Verlag, 2008.Google Scholar
  8. [8]
    Ho T, M’edard M, Shi J, et al. On randomized network cod-Ing[J]. Proceedings of the Annual Allerton Conference on Communication Control and Computing, 2003, 41(1): 11–20.Google Scholar
  9. [9]
    Ho T, M’edard M, Koetter R, et al. A random linear network coding approach to multicast [J]. IEEE Transactions on Information Theory, 2006, 52(10): 4413–4430.CrossRefGoogle Scholar
  10. [10]
    Balli H, Yan X, Zhang Z. On randomized linear network codes and their error correction capabilities[J]. IEEE Transctions on Information Theory, 2009, 55(7): 3148–3160.CrossRefGoogle Scholar
  11. [11]
    Trullols-Cruces O, Barcelo-Ordinas J M, Fiore M. Exact decoding probability under random linear network coding[J]. IEEE Communications Letters, 2011, 15(1): 67–69.CrossRefGoogle Scholar
  12. [12]
    Zhao X B. Notes on ‘exact decoding probability under random linear network coding’ [J]. IEEE Communications Letters, 2012, 16(5): 720–721.CrossRefGoogle Scholar
  13. [13]
    Chiasserini Carla-Fabiana, Viterbo Emanuele, Claudio Casetti. Decoding probability in random linear network codding with packet losses[J]. IEEE Communications Letters, 2013, 17(11): 2128–2131.CrossRefGoogle Scholar
  14. [14]
    Li F, Guo W M. An efficient polynomial time algorithm for robust multicast network code construction[J]. IEEE Communications Letters, 2015, 19(2): 143–146.CrossRefGoogle Scholar
  15. [15]
    Gai Y, Krishnamachari B, Liu M. Online learning for combinatorial network optimization with restless Markovian rewards[ C]// 2012 9th Annual IEEE Communications Society Conference on SECON. Washington D C: IEEE, 2012: 28–36.Google Scholar
  16. [16]
    Gai Y, Krishnamachari B, Jain R. Combinatorial network optimization with unknown variables: Multiarmed bandits with linear rewards and individual observations[J]. IEEE/ACM Transactions on Networking, 2012, 20(5): 1466–1478.CrossRefGoogle Scholar
  17. [17]
    Croitoru M, Croitoru C. Generalised network flows for combinatorial auctions[C]// 2011 IEEE/WIC/ACM International Conference on WI-IAT, 2011, 2: 313–316.Google Scholar
  18. [18]
    Cai N. Network Coding Theory[M]. Boston-Delft: Now Publishers Inc, 2006.Google Scholar
  19. [19]
    Van Lint J H, Wilson R M. A Course in Combinatorics[M]. Cambridge: Cambridge University Press, 2001.CrossRefGoogle Scholar

Copyright information

© Wuhan University and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of InformationXi’an University of Finance and EconomicsXi’an, ShaanxiChina
  2. 2.State Key Laboratory of Integrated Services Networks (ISN)Xidian UniversityXi’an, ShaanxiChina

Personalised recommendations