Skip to main content

LDPC codes generated by conics in the classical projective plane


We construct various classes of low-density parity-check codes using point-line incidence structures in the classical projective plane PG(2,q). Each incidence structure is based on the various classes of points and lines created by the geometry of a conic in the plane. For each class, we prove various properties about dimension and minimum distance. Some arguments involve the geometry of two conics in the plane. As a result, we prove, under mild conditions, the existence of two conics, one entirely internal or external to the other. We conclude with some simulation data to exhibit the effectiveness of our codes.

This is a preview of subscription content, access via your institution.


  1. 1.

    J Cannon C Playoust (1994) An introduction to magma University of Sydney Sydney, Australia

    Google Scholar 

  2. 2.

    RG Gallager (1962) ArticleTitleLow density parity-check codes IRE Trans Infom Theory IT-8 21–28 Occurrence Handle0107.11802 Occurrence Handle136009 Occurrence Handle10.1109/TIT.1962.1057683

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    Hirschfeld JWP (1998) Projective geometries over finite fields, 2nd edn. Oxford University Press

  4. 4.

    Huffman WC, Pless V (2003) Fundamentals of error-correcting codes. Cambridge University Press

  5. 5.

    Kim J-L, Peled UN, Perepelitsa I, Pless V (2002) Explicit construction of families of LDPC codes of girth at least six. In: Proc. 40th Allerton conf. on communication, control and computing, In: Voulgaris PG, Srikant R (eds) Oct. 2–4, 2002, pp 1024–1031

  6. 6.

    Y Kuo S Lin MPC Fossorier (2001) ArticleTitleLow-density parity-check codes based on finite geometries: a rediscovery and new results IEEE Trans Inform Theory 47 IssueID7 2711–2736 Occurrence Handle1872835 Occurrence Handle10.1109/18.959255

    MathSciNet  Article  Google Scholar 

  7. 7.

    DJC MacKay RM Neal (1996) ArticleTitleNear Shannon limit performance of low density parity-check codes Electron Lett 32 IssueID18 1645–1646 Occurrence Handle10.1049/el:19961141

    Article  Google Scholar 

  8. 8.

    KE Mellinger (2004) ArticleTitleLDPC codes and triangle-free line sets Designs Codes Cryptog 32 IssueID1-3 341–350 Occurrence Handle1052.51008 Occurrence Handle2072337 Occurrence Handle10.1023/B:DESI.0000029233.20866.41

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

    Mellinger KE Classes of codes from quadratic surfaces of PG(3,q). Submitted

  10. 10.

    A Passmore J Stovall (2004) ArticleTitleOn codes generated by quadratic surfaces of PG(3,q) Rose-Hulman Inst Technol Undergrad Res J 5 1

    Google Scholar 

  11. 11.

    Rosenthal J, Vontobel PO (2000) Constructions of LDPC codes using Ramanujan graphs and ideas from Margulis. In: Proc. of the 38th annual Allerton conference on communication, control, and computing, pp 248–257

  12. 12.

    B Segre (1955) ArticleTitleOvals in a finite projective plane Can J Math 7 414–416 Occurrence Handle0065.13402 Occurrence Handle71034

    MATH  MathSciNet  Google Scholar 

  13. 13.

    M Sipser DA Spielman (1996) ArticleTitleExpander codes IEEE Trans Inform Theory 42 IssueID6 1710–1722 Occurrence Handle0943.94543 Occurrence Handle1465731 Occurrence Handle10.1109/18.556667

    MATH  MathSciNet  Article  Google Scholar 

  14. 14.

    Storer T (1967) Cyclotomy and difference sets. Markham Publishing Company

  15. 15.

    RM Tanner (1981) ArticleTitleA recursive approach to low complexity codes IEEE Trans Inform Theory IT-27 533–547 Occurrence Handle650686 Occurrence Handle10.1109/TIT.1981.1056404

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Keith E. Mellinger.

Additional information

Communicated by R. Hill

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Droms, S.V., Mellinger, K.E. & Meyer, C. LDPC codes generated by conics in the classical projective plane. Des Codes Crypt 40, 343–356 (2006).

Download citation


  • LDPC code
  • Projective plane
  • Conic

AMS Classification

  • 51E22
  • 94B05