Advertisement

SPC product codes, graphs with cycles and Kostka numbers

  • Sara D. CardellEmail author
  • Joan-Josep Climent
  • Alberto López Martín
Original Paper
  • 17 Downloads

Abstract

The SPC product code is a very popular error correction code with four as its minimum distance. Over the erasure channel, it is supposed to correct up to three erasures. However, this code can correct a higher number of erasures under certain conditions. A codeword of the SPC product code can be represented either by an erasure pattern or by a bipartite graph, where the erasures are represented by an edge. When the erasure contains erasures that cannot be corrected, the corresponding graph contains cycles. In this work we determine the number of strict uncorrectable erasure patterns (bipartite graphs with cycles) for a given size with a fixed number of erasures (edges). Since a bipartite graph can be unequivocally represented by its biadjacency matrix, it is enough to determine the number of non-zero binary matrices whose row and column sum vectors are different from one. At the same time, the number of matrices with prescribed row and column sum vectors can be evaluated in terms of the Kostka numbers associated with Young tableaux.

Keywords

SPC code Erasure channel Erasure pattern Bipartite graph Kostka number 

Mathematics Subject Classification

94B05 05C30 

Notes

References

  1. 1.
    Brualdi, R.A.: Algorithms for constructing (0,1)-matrices with prescribed row and column sum vectors. Discret. Math. 306(23), 3054–3062 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brualdi, R.A.: Introductory combinatorics, 5th edn. Prentice-Hall (Pearson), Englewood Cliffs (2010)Google Scholar
  3. 3.
    Cardell, S.D., Climent, J.J.: An approach to the performance of SPC product codes on the erasure channel. Adv. Math. Commun. 10(1), 11–28 (2016).  https://doi.org/10.3934/amc.2016.10.11 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Diestel, R.: Graph Theory. Springer, New York (2000)zbMATHGoogle Scholar
  5. 5.
    Elias, P.: Coding for noisy channels. IRE Int. Conv. Record, pt. 4, 37–46 (1955)Google Scholar
  6. 6.
    Giblin, P.: Graphs, Surfaces and Homology, 3rd edn. Cambridge University Press, New York (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford Univ Press, Oxford (1975)Google Scholar
  8. 8.
    Hogben, L.: Handbook of Linear Algebra. Chapman and Hall, London (2007)zbMATHGoogle Scholar
  9. 9.
    Knuth, D.E.: Permutations, matrices, and generalized young tableaux. Pac. J. Math. 34(3), 707–727 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kostka, C.: Über den zusammenhang zwischen einigen formen von symmetrischen funktionen. Crelle’s J. 93, 89–123 (1882)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kousa, M.A.: A novel approach for evaluating the performance of SPC product codes under erasure decoding 50(1), 7–11 (2002).  https://doi.org/10.1109/26.975732
  12. 12.
    Kousa, M.A., Mugaibel, A.H.: Cell loss recovery using two-dimensional erasure correction for ATM networks. In: Proceedings of the Seventh International Conference on Telecommunication Systems, pp. 85–89 (1999)Google Scholar
  13. 13.
    Muqaibel, A.: Enhanced upper bound for erasure recovery in SPC product codes. ETRI J. 31(5), 518–524 (2009).  https://doi.org/10.4218/etrij.09.0109.0131 CrossRefGoogle Scholar
  14. 14.
    Rankin, D.M., Gulliver, T.A.: Single parity check product codes 49(8), 1354–1362 (2001).  https://doi.org/10.1109/26.939851
  15. 15.
    Simmons, J.M., Gallager, R.G.: Design of error detection scheme for class C service in ATM. IEEE/ACM Trans. Netw. 2(1), 80–88 (1994).  https://doi.org/10.1109/90.282611 CrossRefGoogle Scholar
  16. 16.
    Stanley, R.: Enumer. Comb., vol. II. Cambridge University Press, Cambridge (1999)Google Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Instituto de Matemática, Estatística e Computação CientíficaUniversity of CampinasCampinasBrazil
  2. 2.Departament de MatemàtiquesUniversitat d’AlacantAlacantSpain
  3. 3.Instituto Nacional de Matemática Pura e AplicadaRio de JaneiroBrazil

Personalised recommendations