Skip to main content
SpringerLink
Log in

Regular LDPC Codes from Semipartial Geometries

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

By considering a class of combinatorial structures, known as semipartial geometries, we define a class of low-density parity-check (LDPC) codes. We derive bounds on minimum distance, rank and girth for the codes from semipartial geometries, and present constructions and performance results for the classes of semipartial geometries which have not previously been proposed for use with iterative decoding.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Gallager, R.G.: Low-density parity-check codes. IRE Trans. Inf. Theory IT-8, 21–28 (1962)

    Article  MathSciNet  Google Scholar 

  2. MacKay, D.J.C., Davey, M.C.: Evaluation of Gallager codes for short block length and high rate applications. In: Marcus, B., Rosenthal, J. (eds.) Codes, Systems and Graphical Models. IMA Volumes in Mathematics and Its Applications, vol. 123, pp. 113–130. Springer, New York (2000)

    Google Scholar 

  3. Lucas, R., Fossorier, M.P.C., Kou, Y., Lin, S.: Iterative decoding of one-step majority logic decodable codes based on belief propagation. IEEE Trans. Commun. 48, 931–937 (2000)

    Article  MATH  Google Scholar 

  4. Kou, Y., Lin, S., Fossorier, M.P.C.: Low-density parity-check codes based on finite geometries: A rediscovery and new results. IEEE Trans. Inf. Theory 47, 2711–2736 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Johnson, S.J., Weller, S.R.: Regular low-density parity-check codes from combinatorial designs. In: Proc. IEEE Information Theory Workshop (1TW2002), pp. 90–92. Cairns, Sept 2001

  6. Johnson, S.J., Weller, S.R.: Resolvable 2-designs for regular low-density parity-check codes. IEEE Trans. Commun. 51, 1413–1419 (2003)

    Article  Google Scholar 

  7. Weller, S.R., Johnson, S.J.: Regular low-density parity-check codes from oval designs. Eur. Trans. Telecommun. 14(5), 399–409 (2003)

    Google Scholar 

  8. Vontobel, P.O., Tanner, R.M.: Construction of codes based on finite generalized quadrangles for iterative decoding. In: Proc. Int. Symp. Information Theory (ISIT’200J), p. 223. Washington, 24–29 June 2001

  9. Johnson, S.J., Weller, S.R.: Codes for iterative decoding from partial geometries. IEEE Trans. Commun. 52, 236–243 (2004)

    Article  Google Scholar 

  10. van Lint, J.H., Wilson, R.M.: A Course in Combinatorics. Cambridge Univ. Press, Cambridge (1992)

    MATH  Google Scholar 

  11. Bose, R.C.: Strongly regular graphs, partial geometries and partial designs. Pac. J. Math. 13, 389–419 (1963)

    MATH  Google Scholar 

  12. Debroey, I., Thas, J.A.: On semipartial geometries. J. Comb. Theory Ser. A 25, 242–250 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  13. Tanner, R.M.: Minimum-distance bounds by graph analysis. IEEE Trans. Inf. Theory 47, 808–821 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cameron, P.J., van Lint, J.H.: Graphs, Codes and Designs. London Math. Soc. Lecture Note Series, vol. 43. Cambridge Univ. Press, Cambridge (1980)

    MATH  Google Scholar 

  15. Buekenhout, F.: Handbook of Incidence Geometry. North-Holland, Amsterdam (1995)

    MATH  Google Scholar 

  16. Brouwer, A.E., van Eijl, C.A.: On the p-rank of strongly regular graphs. Algebra and Combinatorics 1, 72–82 (1992)

    Google Scholar 

  17. Anderson, I.: Combinatorial Designs: Construction Methods, Mathematics and Its Applications. Ellis Horwood, Chichester (1990)

    Google Scholar 

  18. Assmus, E.F. Jr., Key, J.D.: Designs and Their Codes. Cambridge Tracts in Math., vol. 103. Cambridge Univ. Press, Cambridge (1993)

    Google Scholar 

  19. Key, J.D.: Some applications of Magma in designs and codes: Oval designs, Hermitian unitals and generalized Reed-Muller codes. J. Symb. Comput. 31, 37–53 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Thas, J.A.: Semipartial geometries and spreads of classical polar spaces. J. Comb. Theory Ser. A 35, 58–66 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  21. Tits, J.: Ovoïdes et groupes de Suzuki. Arch. Math. 13, 187–198 (1962)

    Article  MathSciNet  Google Scholar 

  22. Coxeter, H.S.M.: Twelve points in PG(5,3) with 95040 self transformations. Proc. R. Soc. Lond. Ser. A 247, 279–293 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pellegrino, G.: Su una interpretazione geometrica dei gruppi M 11 ed M 12 di Mathieu e su alcuni t−(v.k.λ)-disegni deducibili da una (12) 45,3 calotta completa. Atti Semin. Mat. Fis. Univ. Modena 23, 103–117 (1975)

    MathSciNet  Google Scholar 

  24. Hill, R.: On the largest size cap in S 5,3. Atti. Accad. Naz. Lincei Rend. 54, 378–384 (1973)

    Google Scholar 

  25. Hill, R.: Caps and groups. In: Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo II. Atti dei Convegni Lincei, vol. 17, pp. 389–394. Accad. Naz. Lincei, Rome (1976)

    Google Scholar 

  26. MacKay, D.J.C.: Good error-correcting codes based on very sparse matrices. IEEE Trans. Inf. Theory 45, 399–431 (1999)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Math. and Phys., Qingdao University of Science and Technology, Qingdao, 266042, People’s Republic of China

    Xiuli Li

  2. Institute of Wireless Communication Technology, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Chen Zhang

  3. College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai, 201620, People’s Republic of China

    Jun Shen

Authors
  1. Xiuli Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chen Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jun Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiuli Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, X., Zhang, C. & Shen, J. Regular LDPC Codes from Semipartial Geometries. Acta Appl Math 102, 25–35 (2008). https://doi.org/10.1007/s10440-007-9186-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-007-9186-y

Keywords

Search

Navigation