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LDPC Codes from \(\mu \)-Geodetic Graphs Obtained from Block Designs

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Abstract

In this paper we study low-density parity-check (LDPC) codes spanned by rows of the adjacency matrices of \(\mu \)-geodetic graphs obtained from 2-\((v,k,\lambda )\) designs. This construction can be applied to any 2-\((v,k,\lambda )\) design, but in this paper we focus on the LDPC codes from \(\mu \)-geodetic graphs obtained from 2-\((v,3,\lambda )\) designs since their Tanner graphs are free of 4-cycles. We analyse some properties of the constructed LDPC codes and discuss code length, dimension and minimum distance. Further, we discuss absorbing sets of these LDPC codes and give an expression for the expectation of a syndrome weight of the constructed LDPC codes. Information on the constructed LDPC codes, such as their parameters, is presented as well.

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References

  1. Ammar, B., Honary, B., Kou, Y., Xu, J., Lin, S.: Construction of low density parity-check codes based on balanced incomplete block designs. IEEE Trans. Inform. Theory 50, 1257–1267 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beth, T., Jungnickel, D., Lenz, H.: Design Theory, vol. I. Cambridge University Press, Cambridge, England (1999)

    Book  MATH  Google Scholar 

  3. Crnković, D., Dumičić Danilović, D., Rukavina, S.: Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms. J. Algebra Comb. Discrete Struct. Appl. 3, 145–154 (2016)

    MathSciNet  Google Scholar 

  4. Crnković, D., Rukavina, S., Simčić, L.: On triplanes of order twelve admitting an automorphism of order six and their binary and ternary codes. Util. Math. 103, 23–40 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Di, C., Proietti, D., Telatar, I.E., Richardson, T.J., Urbanke, R.L.: Finite-length analysis of low-density parity-check codes on the binary erasure channel. IEEE Trans. Inform. Theory 48, 1570–1579 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Diao, Q., Tai, Y.Y., Lin, S., Abdel-Ghaffar, K.: Trapping Set Structure of LDPC Codes on Finite Geometries. IEEE Int. Symp. on Inf. Theory, pp 3088–3092 (2013)

  7. Diestel, R.: Graph theory, Fourth edition, Graduate Texts in Mathematics 173. Springer, Heidelberg (2010)

    Google Scholar 

  8. Falsafain, H., Esmaeili, M.: A new construction of structured binary regular LDPC codes based on steiner systems with parameter \(t>2\). IEEE Trans. Commun. 60, 74–80 (2011)

    Article  Google Scholar 

  9. Fujiwara, Y.: Instantaneous quantum channel estimation during quantum information processing. (2014). arXiv:1405.6267

  10. Fujiwara, Y., Gruner, A., Vandendriessche, P.: High-rate quantum low-density parity-check codes assisted by reliable qubits. IEEE Trans. Inform. Theory 61, 1860–1878 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gallager, R.G.: Low-Density Parity-Check Codes. MIT Press, Cambridge (1963)

    MATH  Google Scholar 

  12. Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de, Accessed 29 Jan 2016

  13. Hamada, N.: On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error correcting codes. Hiroshima Math. J. 3, 153–226 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ivanov, F., Zyablov, V.: LDPC codes based on Steiner quadruple systems and permutation matrices. In: Fourteenth International Workshop on Algebraic and Combinatorial Coding Theory, Svetlogorsk, Russia pp. 175–180 (2014)

  15. Johnson, S.J., Weller, S.R.: Construction of low-density parity-check codes from Kirkman triple systems. IEEE Global Commun. Conf. 2, 970–974 (2001)

    Google Scholar 

  16. Kou, Y., Lin, S., Fossorier, M.P.C.: Low density parity check codes based on finite geometries: a rediscovery and new results. IEEE Trans. Inform. Theory 47, 2711–2736 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lechner, G., Pacher, C.: Estimating channel parameters from the syndrome of a linear code. IEEE Commun. Lett. 17, 2148–2151 (2013)

    Article  Google Scholar 

  18. MacKay, D.J.C., Neal, R.M.: Near Shannon limit performance of low density parity check codes. Electron. Lett. 32, 457–458 (1997)

    Article  Google Scholar 

  19. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes, 12th edn. North-Holland, Amsterdam (2006)

    MATH  Google Scholar 

  20. Moura, J.M.F., Lu, J., Zhang, H.: Structured low-density parity-check codes. IEEE Signal Process. Mag. 21, 42–55 (2004)

    Article  Google Scholar 

  21. Pacher, C., Grabenweger, P., Simos, D.E.: Weight distribution of the syndrome of linear codes and connections to combinatorial designs. Inform. Theory (ISIT), pp. 3038–3042 (2016)

  22. Schlegel, C.: On the dynamics of the error floor behavior in (regular) LDPC codes. IEEE Trans. Inform. Theory 56, 3248–3264 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Srinivasan, N., Opatrny, J., Alagar, V.S.: Construction of geodetic and bigeodetic blocks of connectivity \(k\ge 3\) and their relation to block designs. Ars Combin. 24, 101–114 (1987)

    MathSciNet  MATH  Google Scholar 

  24. Stinson, D.R.: Combinatorial Designs: Construction and Analysis. Springer, New York (2004)

    MATH  Google Scholar 

  25. Toto-Zarasoa, V., Roumy, A., Guillemot, C.: Maximum likelihood BSC parameter estimation for the Slepian–Wolf problem. IEEE Commun. Lett. 15, 232–234 (2011)

    Article  Google Scholar 

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Acknowledgements

This work has been fully supported by Croatian Science Foundation under the project 6732. The authors would like to thank the anonymous referees for the helpful comments and suggestions.

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  1. Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000, Rijeka, Croatia

    Dean Crnković, Sanja Rukavina & Marina Šimac

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  1. Dean Crnković
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  2. Sanja Rukavina
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  3. Marina Šimac
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Correspondence to Marina Šimac.

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Crnković, D., Rukavina, S. & Šimac, M. LDPC Codes from \(\mu \)-Geodetic Graphs Obtained from Block Designs. Graphs and Combinatorics 35, 451–469 (2019). https://doi.org/10.1007/s00373-019-02007-4

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