The weight distribution of the functional codes defined by forms of degree 2 on Hermitian surfaces

Frédéric A. B. Edoukou[1]

  • [1] CNRS, Institut de Mathématiques de Luminy Luminy case 907 13288 Marseille Cedex 9 - France

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 1, page 131-143
  • ISSN: 1246-7405

Abstract

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We study the functional codes C 2 ( X ) defined on a projective algebraic variety X , in the case where X ⊂ ℙ 3 ( 𝔽 q ) is a non-degenerate Hermitian surface. We first give some bounds for # X Z ( 𝒬 ) ( 𝔽 q ) , which are better than the ones known. We compute the number of codewords reaching the second weight. We also estimate the third weight, show the geometrical structure of the codewords reaching this third weight and compute their number. The paper ends with a conjecture on the fourth weight and the fifth weight of the code C 2 ( X ) .

How to cite

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Edoukou, Frédéric A. B.. "The weight distribution of the functional codes defined by forms of degree 2 on Hermitian surfaces." Journal de Théorie des Nombres de Bordeaux 21.1 (2009): 131-143. <http://eudml.org/doc/10866>.

@article{Edoukou2009,
abstract = {We study the functional codes $C_2(X)$ defined on a projective algebraic variety $X$, in the case where $X \subset \{\mathbb\{P\}\}^3(\mathbb\{F\}_\{q\})$ is a non-degenerate Hermitian surface. We first give some bounds for $\# X_\{ Z(\mathcal\{Q\})\}(\mathbb\{F\}_\{q\} )$, which are better than the ones known. We compute the number of codewords reaching the second weight. We also estimate the third weight, show the geometrical structure of the codewords reaching this third weight and compute their number. The paper ends with a conjecture on the fourth weight and the fifth weight of the code $C_2(X)$.},
affiliation = {CNRS, Institut de Mathématiques de Luminy Luminy case 907 13288 Marseille Cedex 9 - France},
author = {Edoukou, Frédéric A. B.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {functional codes; minimum distance; weight distribution},
language = {eng},
number = {1},
pages = {131-143},
publisher = {Université Bordeaux 1},
title = {The weight distribution of the functional codes defined by forms of degree 2 on Hermitian surfaces},
url = {http://eudml.org/doc/10866},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Edoukou, Frédéric A. B.
TI - The weight distribution of the functional codes defined by forms of degree 2 on Hermitian surfaces
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 1
SP - 131
EP - 143
AB - We study the functional codes $C_2(X)$ defined on a projective algebraic variety $X$, in the case where $X \subset {\mathbb{P}}^3(\mathbb{F}_{q})$ is a non-degenerate Hermitian surface. We first give some bounds for $\# X_{ Z(\mathcal{Q})}(\mathbb{F}_{q} )$, which are better than the ones known. We compute the number of codewords reaching the second weight. We also estimate the third weight, show the geometrical structure of the codewords reaching this third weight and compute their number. The paper ends with a conjecture on the fourth weight and the fifth weight of the code $C_2(X)$.
LA - eng
KW - functional codes; minimum distance; weight distribution
UR - http://eudml.org/doc/10866
ER -

References

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  5. R. Hartshorne, Algebraic Geometry. Graduate texts in mathematics 52, Springer-Verlag, 1977. Zbl0367.14001MR463157
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  7. J. W. P. Hirschfeld, Finite projective spaces of three dimensions. Clarendon press. Oxford, 1985. Zbl0574.51001MR840877
  8. G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties. In “Arithmetic, Geometry, and Coding Theory”. (Luminy, France, June 17-21, 1993), Walter de Gruyter, Berlin-New York, 1996, 77–104. Zbl0876.94046MR1394928
  9. I. R. Shafarevich, Basic algebraic geometry 1. Springer-Verlag, 1994. Zbl0797.14001MR1328833
  10. A. B. Sørensen, Rational points on hypersurfaces, Reed-Muller codes and algebraic-geometric codes. Ph. D. Thesis, Aarhus, Denmark, 1991. 
  11. P. Spurr, Linear codes over G F ( 4 ) . Master’s Thesis, University of North Carolina at Chapell Hill, USA, 1986. 

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