Advertisement

On LDPC Codes Based on Families of Expanding Graphs of Increasing Girth without Edge-Transitive Automorphism Groups

  • Monika Polak
  • Vasyl Ustimenko
Part of the Communications in Computer and Information Science book series (CCIS, volume 448)

Abstract

We introduce new examples of Low Density Parity Check codes connected with the new families of regular graphs of bounded degree and increasing girth. Some new codes have an evident advantage in comparison with the D(n,q) based codes [9]. The new graphs are not edge transitive. So, they are not isomorphic to the Cayley graphs or those from the D(n,q) family, [14]. We use computer simulation to investigate spectral properties of graphs used for the construction of new codes. The experiment demonstrates existence of large spectral gaps in the case of each graph. We conjecture the existence of infinite families of Ramanujan graphs and expanders of bounded degree,existence of strongly Ramanujan graphs of unbounded degree. The lists of eigenvalues can be used for various practical applications of expanding graphs (Coding Theory, Networking, Image Processing). We show that new graphs can be used as a source of lists of cospectral pairs of graphs of bounded or unbounded degree.

Keywords

expanding graphs Ramanujun graphs LDPC codes families of graphs of increasing girth spectral gap cospectral graphs 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N.: Eigenvalues, geometric expanders, sorting in rounds and Ramsey theory. Combinatorica 6(3), 207–219 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Biggs, N.L.: Algebraic Graph Theory, 2nd edn. Cambridge University Press (1993)Google Scholar
  3. 3.
    Biggs, N.L.: Graphs with large girth. Ars Combin. In: Eleventh British Combinatorial Conference, London, vol. 25(C), pp. 73–80 (1988)Google Scholar
  4. 4.
    Bollobas, B.: Extremal Graph Theory. Academic Press, London (1978)zbMATHGoogle Scholar
  5. 5.
    Chiu, P.: Cubic Ramanujan graphs. Combinatorica 12(3), 275–285 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Erdős, P.: Graph theory and probability. Canad. Math. Monthly 11, 34–38 (1959)CrossRefGoogle Scholar
  7. 7.
    Gallager, R.G.: Low-Density Parity-Checks Codes. Monograph. M.I.T. Press (1963)Google Scholar
  8. 8.
    Guinand, P., Lodge, J.: Graph theoretic construction of generalized product codes. In: IEEE International Symposium on Information Theory, ISIT 1997, Ulm, Germany, June 29-July 4, p. 111 (1997)Google Scholar
  9. 9.
    Guinand, P., Lodge, J.: Tanner type codes arising from large girth graphs. In: Canadian Workshop on Information Theory CWIT 1997, Toronto, Ontario, Canada, pp. 5–7 (1997)Google Scholar
  10. 10.
    Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.) 43(4), 439–561 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Imrich, W.: Explicit construction of regular graphs without small cycles. Combinatorica 4(1), 53–59 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Lazebnik, F., Ustimenko, V., Woldar, A.J.: Polarities and 2k-cycle-free graphs. Discrete Mathematics 197-198, 503–513 (1999)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Lazebnik, F., Ustimenko, V., Woldar, A.J.: A characterization of the components of the graphs D(k,q). Discrete Mathematics 157, 271–283 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Lazebnik, F., Ustimenko, V., Woldar, A.J.: A new series of dense graphs of high girth. Bulletin (New Series) of the AMS 32(1), 73–79 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Lazebnik, F., Ustimenko, V., Woldar, A.J.: Polarities and 2k-cycle-free graphs. Discrete Mathematics 197-198, 503–513 (1999)Google Scholar
  16. 16.
    Lubotsky, A., Philips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Luby, M.G., Mitzenmacher, M., Shokrollahi, M.A., Spielman, D.A.: Improved Low-Density Parity-Check Codes Using Irregular Graphs and Belief Propagation. In: ISIT 1998-IEEE International Symposium of Information Theory, Cambridge, USA, p. 171 (1998)Google Scholar
  18. 18.
    MacKay, D.J.C., Neal, R.M.: Good Codes Based on Very Sparse Matrices. In: Boyd, C. (ed.) Cryptography and Coding 1995. LNCS, vol. 1025, pp. 100–111. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  19. 19.
    MacKay, D., Postol, M.: Weakness of Margulis and Ramanujan Margulis Low Dencity Parity Check Codes. Electronic Notes in Theoretical Computer Science, vol. 74 (2003)Google Scholar
  20. 20.
    Margulis, G.A.: Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi Informatsii 24(1), 51–60 (1988)MathSciNetGoogle Scholar
  21. 21.
    Margulis, G.A.: Explicit construction of graphs without short cycles and low density codes. Combinatorica 2, 1–78 (1982)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Morgenstern, M.: Existence and explicit constructions of q + 1-regular Ramanujan graphs for every prime power q. J. Combin. Theory Ser. B 62(1), 44–62 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Polak, M., Ustimenko, V.: On LDPC codes Corresponding to Infinite Family of Graphs A(n,K). In: Proceedings of Federated Conference on Computer Science and Informations Systems, Wrocław, Poland, September 9-12, pp. 567–570 (2012)Google Scholar
  24. 24.
    Polak, M., Ustimenko, V.: Appendix for article On LDPC codes based on families of expanding graphs of increasing girth without edge-transitive automorphism groups. University of Maria Curie Skłodowska (2014), http://umcs.pl/pl/zaklad-algebry-i-matematyki-dyskretnej,1336.htm
  25. 25.
    Richardson, T.J., Urbanke, R.L.: The Capacity of Low-Density Parity Check Codes Under Message-Passing Decoding. IEEE Transaction on Informarion Theory 47(2), 599–618 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Akos, S.: Large Families of Cospectral Graphs. Designs, Codes and Cryptography 21(1-3), 205–208 (2000)zbMATHGoogle Scholar
  27. 27.
    Sipser, M., Spielman, D.A.: Expander codes. IEEE Trans. on Info. Theory 42(6), 1710 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Shannon, C.E., Warren, W.: The Mathematical Theory of Communication. University of Illinois Press (1963)Google Scholar
  29. 29.
    Tanner, R.M.: A recursive approach to low density codes. IEEE Transactions on Information Theory IT 27(5), 533–547 (1984)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Ustimenko, V., Woldar, A.: Extremal properties of regular and affine generalized polygons as tactical configurations. Eur. J. Combinator. 24, 99 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Ustimenko, V.: On linguistic dynamical systems, families of graphs of large girth, and cryptography. Zapiski Nauchnykh Seminarov POMI 326, 214–234 (2005)zbMATHGoogle Scholar
  32. 32.
    Ustimenko, V.: On the K-theory of graph based dynamical systems and its applications. Dopovidi Natsional’noi Akademii nauk Ukrainy 8, 44–51 (2013)Google Scholar
  33. 33.
    Ustimenko, V.: On extremal graph theory and symbolic computations. Dopovidi Natsional’noi Akademii nauk Ukrainy 2, 42–49 (2013)Google Scholar
  34. 34.
    Romańczuk, U., Ustimenko, V.: On the key exchange protocol with new cubical maps based on graphs. Annales UMCS Informaticea XI, 11–29 (2011)Google Scholar
  35. 35.
    Ustimenko, V., Romańczuk, U.: On Extremal Graph Theory, Explicit algebraic constructions of extremal graphs and corresponding Turing encryption machines. In: Yang, X.-S. (ed.) Artificial Intelligence, Evolutionary Computing and Metaheuristics. SCI, vol. 427, pp. 257–285. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  36. 36.
    Weiss, A.: Girths of bipartite sextet graphs. Combinatorica 4(2-3), 241–245 (1984)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Monika Polak
    • 1
  • Vasyl Ustimenko
    • 1
  1. 1.Institute of MathematicsMaria Curie-Sklodowska UniversityLublinPoland

Personalised recommendations