Advertisement

Designs, Codes and Cryptography

, Volume 71, Issue 1, pp 21–46 | Cite as

On the functional codes defined by quadrics and Hermitian varieties

  • D. Bartoli
  • M. De BoeckEmail author
  • S. Fanali
  • L. Storme
Article

Abstract

In recent years, functional codes have received much attention. In his PhD thesis, F.A.B. Edoukou investigated various functional codes linked to quadrics and Hermitian varieties defined in finite projective spaces (Edoukou, PhD Thesis, 2007). This work was continued in (Edoukou et al., Des Codes Cryptogr 56:219–233, 2010; Edoukou et al., J Pure Appl Algebr 214:1729–1739, 2010; Hallez and Storme, Finite Fields Appl 16:27–35, 2010), where the results of the thesis were improved and extended. In particular, Hallez and Storme investigated the functional codes \({C_2(\mathcal{H})}\), with \({\mathcal{H}}\) a non-singular Hermitian variety in PG(N, q 2). The codewords of this code are defined by evaluating the points of \({\mathcal{H}}\) in the quadratic polynomials defined over \({\mathbb{F}_{q^2}}\). We now present the similar results for the functional code \({C_{Herm}(\mathcal{Q})}\). The codewords of this code are defined by evaluating the points of a non-singular quadric \({\mathcal{Q}}\) in PG(N, q 2) in the polynomials defining the Hermitian varieties of PG(N, q 2).

Keywords

Functional codes Quadrics Hermitian varieties code divisor 

Mathematics Subject Classification

05B25 51E20 94B05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Edoukou F.A.B.: Codes correcteurs d’erreurs construits à partir des variétés algébriques. PhD Thesis, Université de la Méditerranée Aix-Marseille II (2007).Google Scholar
  2. 2.
    Edoukou F.A.B., Hallez A., Rodier F., Storme L.: On the small weight codewords of the functional codes C herm(X), X a non-singular Hermitian variety. Des. Codes Cryptogr. 56, 219–233 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Edoukou F.A.B., Hallez A., Rodier F., Storme L.: A study of intersections of quadrics having applications on the small weight codewords of the functional codes C 2(Q), Q a non-singular quadric. J. Pure Appl. Algebr. 214, 1729–1739 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Edoukou F.A.B., Ling S., Xing C.: Structure of functional codes defined on non-degenerate Hermitian varieties. J. Combin. Theory A 118, 2436–2444 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Hallez A., Storme L.: Functional codes arising from quadric intersection with Hermitian varieties. Finite Fields Appl. 16, 27–35 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Hirschfeld J.W.P., Thas J.A.: General Galois Geometries. Mathematical Monographs. Oxford University Press, Oxford (1991)Google Scholar
  7. 7.
    Katz N.M.: On a theorem of Ax. Am. J. Math. 93, 485–499 (1971)CrossRefzbMATHGoogle Scholar
  8. 8.
    Lachaud, G.: Number of points of plane sections and linear codes defined on algebraic varieties. In: Arithmetic, Geometry, and Coding Theory. (Luminy, France, 1993), pp 77–104. Walter De Gruyter, Berlin (1996).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly
  2. 2.Department of MathematicsGhent UniversityGhentBelgium
  3. 3.Dipartimento di Matematica e InformaticaUniversità della BasilicataPotenzaItaly
  4. 4.Department of MathematicsGhent UniversityGhentBelgium

Personalised recommendations