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Designs, Codes and Cryptography

, Volume 72, Issue 2, pp 381–404 | Cite as

2-Ranks of incidence matrices associated with conics in finite projective planes

  • Megan Adams
  • Junhua Wu
Article

Abstract

In this article, we investigate geometric properties of the secant-internal neighbors of internal points and the passant-external neighbors of external points in classical finite projective planes; we calculate the \(2\)-ranks of the incidence matrices of internal points versus their secant-internal neighbors and external points versus their passant-external neighbors using a combination of techniques from both finite geometry and linear algebra.

Keywords

Conic Incidence matrix Low-density parity-check code Module \(2\)-Rank. 

Mathematics Subject Classification (1991)

51E20 05B20 

Notes

Acknowledgments

This research was supported in part by NSF HBCU-UP Grant Award \(0929257\) at Lane College.

References

  1. 1.
    Assmus E.F., Key J.D.: Designs and Their Codes. Cambridge University Press, New York (1992).Google Scholar
  2. 2.
    Droms S., Mellinger K.E., Meyer C.: LDPC codes generated by conics in the classical projective plane. Des. Codes Cryptogr. 40, 343–356 (2006).Google Scholar
  3. 3.
    Dye R.H.: Hexagons, conics, \(A_5\) and \({\text{ P}SL}_2(K)\). J. Lond. Math. Soc. 44(2), 270–286 (1991).Google Scholar
  4. 4.
    Hirschfeld J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Oxford University Press, Oxford (1998).Google Scholar
  5. 5.
    Hughes D.R., Piper F.C.: Projective Planes, Graduate Texts in Mathematics, 6th edn. Springer, New York.Google Scholar
  6. 6.
    Madison A.L., Wu J.: On binary codes from conics in PG\((2, q)\). Eur. J. Comb. 33, 33–48 (2012).Google Scholar
  7. 7.
    Sin P., Wu J., Xiang Q.: Dimensions of some binary codes arising from a conic in PG\((2, q)\). J. Comb. Theory Ser. A 118, 853–878 (2011).Google Scholar
  8. 8.
    Wu J.: Proofs of two conjectures on the dimensions of binary codes. Des. Codes Cryptogr (in press).Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of MathematicsLane CollegeJacksonUSA

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