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Absorbing sets of codes from finite geometries

  • Allison Beemer
  • Kathryn HaymakerEmail author
  • Christine A. Kelley
Article
  • 21 Downloads
Part of the following topical collections:
  1. Special Issue on Coding Theory and Applications

Abstract

We examine the presence of absorbing sets, fully absorbing sets, and elementary absorbing sets in low-density parity-check (LDPC) codes arising from certain classes of finite geometries. In particular, we prove the parameters of the smallest absorbing sets for finite geometry codes using a tree-based argument. Moreover, we obtain the parameters of the smallest absorbing sets for a special class of codes whose graphs are d-left-regular with girth g = 6 or g = 8.

Keywords

Absorbing sets Finite geometry LDPC codes Tree bounds 

Notes

Acknowledgements

K. Haymaker would like to thank Pascal Vontobel for noting an error in Lemma 2 of [1], which was helpful in the preparation of this paper.

References

  1. 1.
    Haymaker, K.: Absorbing set analysis of codes from affine planes. In: Barbero, Á. I., Skachek, V., Ytrehus, Ø. (eds.) International Castle Meeting on Coding Theory and Applications 2017, Lecture Notes in Computer Science 10495, pp. 154–162. Springer, Cham (2017)Google Scholar
  2. 2.
    Kou, Y., Lin, S., Fossorier, M.: Construction of low density parity check codes: a geometric approach. In: Proceedings of the 2nd IEEE International Symposium on Turbo Codes and Related Topics, pp. 137–140 (2000)Google Scholar
  3. 3.
    Kou, Y., Lin, S., Fossorier, M.: Low-density parity-check codes based on finite geometries: a rediscovery and new results. IEEE Trans. Inf. Theory 47(7), 2711–2736 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kelley, C.A., Sridhara, D., Rosenthal, J.: Tree-based construction of LDPC codes having good pseudocodeword weights. IEEE Trans. Inf. Theory 53(4), 1460–1478 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Xia, S.T., Fu, F.W.: On the stopping distance of finite geometry LDPC codes. IEEE Commun. Lett. 10(5), 381–383 (2006)CrossRefGoogle Scholar
  6. 6.
    Smarandache, R., Vontobel, P. O.: Pseudo-codeword analysis of Tanner graphs from projective and Euclidean planes. IEEE Trans. Inf. Theory 53(7), 2376–2393 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Landner, S., Milenkovic, O.: Algorithmic and combinatorial analysis of trapping sets in structured LDPC codes. In: Proceedings of the IEEE Wireless Networks Communications and Mobile Computing, pp. 630–635 (2005)Google Scholar
  8. 8.
    Diao, Q., Tai, Y.Y., Lin, S., Abdel-Ghaffar, K.: Trapping set structure of LDPC codes on finite geometries. In: Proceedings of the IEEE Information Theory and Applications Workshop (ITA), pp. 1–8 (2013)Google Scholar
  9. 9.
    Liu, H., Li, Y., Ma, L., Chen, J.: On the smallest absorbing sets of LDPC codes from finite planes. IEEE Trans. Inf. Theory 58(6), 4014–4020 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Koetter, R., Vontobel, P.O.: Graph covers and iterative decoding of finite-length codes. In: Proceedings of the 3rd International Symposium on Turbo Codes and Related Topics, Brest, France, pp. 75–82 (2003)Google Scholar
  11. 11.
    Dolecek, L., Zhang, Z., Anantharam, V., Wainwright, M., Nikolic, B.: Analysis of absorbing sets for array-based LDPC codes. In: Proceedings of the IEEE International Conference on Communications, pp. 6261–6268 (2007)Google Scholar
  12. 12.
    Richardson, T.: Error floors of LDPC codes. In: Proceedings of the Annual Allerton Conference on Communications Control, and Computing, vol. 41, pp. 1426–1435 (2003)Google Scholar
  13. 13.
    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. Inf. Theory 48(6), 1570–1579 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dolecek, L., Lee, P., Zhang, Z., Anantharam, V., Nikolic, B., Wainwright, M.: Predicting error floors of structured LDPC codes: deterministic bounds and estimates. IEEE J. Sel. Areas Commun. 27(6), 908–917 (2009)CrossRefGoogle Scholar
  15. 15.
    Zhang, S., Schlegel, C.: Controlling the error floor in LDPC decoding. IEEE Trans. Commun. 61(9), 3566–3575 (2013)CrossRefGoogle Scholar
  16. 16.
    Hatami, H., Mitchell, D.G.M., Costello, D.J., Fuja, T.: Performance bounds for quantized LDPC decoders based on absorbing sets. In: Proceedings of the IEEE International Symposium on Information Theory, Barcelona, Spain, pp. 2539–2543 (2016)Google Scholar
  17. 17.
    Tanner, R. M.: A recursive approach to low-complexity codes. IEEE Trans. Inf. Theory 27(5), 533–547 (1981)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dolecek, L.: On absorbing sets of structured sparse graph codes. In: Proceedings of the IEEE Information Theory and Applications Workshop (ITA), pp. 1–5 (2010)Google Scholar
  19. 19.
    Schwartz, M., Vardy, A.: On the stopping distance and the stopping redundancy of codes. IEEE Trans. Inf. Theory 52(3), 922–932 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Feldman, J., Wainwright, M.J., Karger, D.R.: Using linear programming to decode binary linear codes. IEEE Trans. Inf. Theory 51(3), 954–972 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Batten, L. M.: Combinatorics of Finite Geometries, 2nd edn. Cambridge University Press, Cambridge (1997)Google Scholar
  22. 22.
    Tang, H., Xu, J., Lin, S., Abdel-Ghaffar, K.A.S.: Codes on finite geometries. IEEE Trans. Inf. Theory 51(2), 572–596 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Amiri, B., Lin, C.W., Dolecek, L.: Asymptotic distribution of absorbing sets and fully absorbing sets for regular sparse code ensembles. IEEE Trans. Commun. 61 (2), 455–464 (2013)CrossRefGoogle Scholar
  24. 24.
    Kyung, G.B., Wang, C.C.: Finding the exhaustive list of small fully absorbing sets and designing the corresponding low error-floor decoder. IEEE Trans. Commun. 60(6), 1487–1498 (2012)CrossRefGoogle Scholar
  25. 25.
    Johnson, S.J., Weller, S.R.: Codes for iterative decoding from partial geometries. IEEE Trans. Commun. 52(2), 236–243 (2004)CrossRefGoogle Scholar
  26. 26.
    Tanner, R.M.: Explicit concentrators from generalized N-gons. SIAM J. Algebr. Discret. Methods 5(3), 287–293 (1984)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hirschfeld, J.W., Storme, L.: The packing problem in statistics, coding theory and finite projective spaces: update 2001. In: Finite Geometries, pp. 201–246. Springer, Boston (2001)Google Scholar
  28. 28.
    Ball, S.: Finite Geometry and Combinatorial Applications. London Mathematical Society Student Texts Vol. 82. Cambridge University Press, Cambridge (2015)CrossRefGoogle Scholar
  29. 29.
    Bierbrauer, J., Edel, Y.: Large caps in projective Galois spaces. In: De Beule, M., Storme, L (eds.) Current Research Topics in Galois Geometry, pp. 85–102. Nova Science Publishers (2011)Google Scholar
  30. 30.
    Beemer, A., Habib, S., Kelley, C.A., Kliewer, J.: A generalized algebraic approach to optimizing SC-LDPC codes. In: Proceedings of the Annual Allerton Conference on Communication Control, and Computing, vol. 55, pp. 672–679 (2017)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Arizona State UniversityTempeUSA
  2. 2.Villanova UniversityVillanovaUSA
  3. 3.University of Nebraska-LincolnLincolnUSA

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