Skip to main content
SpringerLink
Log in

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

  • Conference paper
Cryptography and Security Systems (CSS 2014)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 448))

Included in the following conference series:

  • 1061 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alon, N.: Eigenvalues, geometric expanders, sorting in rounds and Ramsey theory. Combinatorica 6(3), 207–219 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Biggs, N.L.: Algebraic Graph Theory, 2nd edn. Cambridge University Press (1993)

    Google Scholar 

  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. Bollobas, B.: Extremal Graph Theory. Academic Press, London (1978)

    MATH  Google Scholar 

  5. Chiu, P.: Cubic Ramanujan graphs. Combinatorica 12(3), 275–285 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Erdős, P.: Graph theory and probability. Canad. Math. Monthly 11, 34–38 (1959)

    Article  Google Scholar 

  7. Gallager, R.G.: Low-Density Parity-Checks Codes. Monograph. M.I.T. Press (1963)

    Google Scholar 

  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. 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. Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.) 43(4), 439–561 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Imrich, W.: Explicit construction of regular graphs without small cycles. Combinatorica 4(1), 53–59 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lazebnik, F., Ustimenko, V., Woldar, A.J.: Polarities and 2k-cycle-free graphs. Discrete Mathematics 197-198, 503–513 (1999)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lazebnik, F., Ustimenko, V., Woldar, A.J.: Polarities and 2k-cycle-free graphs. Discrete Mathematics 197-198, 503–513 (1999)

    Google Scholar 

  16. Lubotsky, A., Philips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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. 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)

    MathSciNet  Google Scholar 

  21. Margulis, G.A.: Explicit construction of graphs without short cycles and low density codes. Combinatorica 2, 1–78 (1982)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  26. Akos, S.: Large Families of Cospectral Graphs. Designs, Codes and Cryptography 21(1-3), 205–208 (2000)

    MATH  Google Scholar 

  27. Sipser, M., Spielman, D.A.: Expander codes. IEEE Trans. on Info. Theory 42(6), 1710 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  28. Shannon, C.E., Warren, W.: The Mathematical Theory of Communication. University of Illinois Press (1963)

    Google Scholar 

  29. Tanner, R.M.: A recursive approach to low density codes. IEEE Transactions on Information Theory IT 27(5), 533–547 (1984)

    Article  MathSciNet  Google Scholar 

  30. Ustimenko, V., Woldar, A.: Extremal properties of regular and affine generalized polygons as tactical configurations. Eur. J. Combinator. 24, 99 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Ustimenko, V.: On linguistic dynamical systems, families of graphs of large girth, and cryptography. Zapiski Nauchnykh Seminarov POMI 326, 214–234 (2005)

    MATH  Google Scholar 

  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. Ustimenko, V.: On extremal graph theory and symbolic computations. Dopovidi Natsional’noi Akademii nauk Ukrainy 2, 42–49 (2013)

    Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  36. Weiss, A.: Girths of bipartite sextet graphs. Combinatorica 4(2-3), 241–245 (1984)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Maria Curie-Sklodowska University, pl. M. Curie-Sklodowskiej 5, 20-031, Lublin, Poland

    Monika Polak & Vasyl Ustimenko

Authors
  1. Monika Polak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vasyl Ustimenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Insitute of Telecommunications, Warsaw University of Technology, ul. Nowowiejska 15/19, 00-665, Warsaw, Poland

    Zbigniew Kotulski

  2. Institute of Computer Science, Maria Curie-Skłodowska in Lublin, ul. Plac Marii Curie-Skłodowskiej 5, 20-031 Lublin and Polish-Japanese Institute of Information Technology, Koszykowa 8, Poland

    Bogdan Księżopolski

  3. Institute of Computer Science, Maria Curie-Skłodowska in Lublin, ul. Plac Marii Curie-Skłodowskiej 5, 20-031, Lublin, Poland

    Katarzyna Mazur

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Polak, M., Ustimenko, V. (2014). On LDPC Codes Based on Families of Expanding Graphs of Increasing Girth without Edge-Transitive Automorphism Groups. In: Kotulski, Z., Księżopolski, B., Mazur, K. (eds) Cryptography and Security Systems. CSS 2014. Communications in Computer and Information Science, vol 448. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44893-9_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-44893-9_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44892-2

  • Online ISBN: 978-3-662-44893-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation