Differential approach for the study of duals of algebraic-geometric codes on surfaces

Alain Couvreur[1]

  • [1] INRIA Saclay, Projet Tanc École polytechnique Laboratoire d’informatique LIX, UMR 7161 91128 Palaiseau Cedex, France

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 1, page 95-120
  • ISSN: 1246-7405

Abstract

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The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this description is less trivial, it can be regarded as a natural extension to surfaces of the result asserting that the dual of a functional code C L ( D , G ) on a curve is the differential code C Ω ( D , G ) . We study the parameters of such codes and state a lower bound for their minimum distance. Using this bound, one can study some examples of codes on surfaces, and in particular surfaces with Picard number 1 like elliptic quadrics or some particular cubic surfaces. The parameters of some of the studied codes reach those of the best known codes up to now.

How to cite

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Couvreur, Alain. "Differential approach for the study of duals of algebraic-geometric codes on surfaces." Journal de Théorie des Nombres de Bordeaux 23.1 (2011): 95-120. <http://eudml.org/doc/219685>.

@article{Couvreur2011,
abstract = {The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this description is less trivial, it can be regarded as a natural extension to surfaces of the result asserting that the dual of a functional code $C_L (D,G)$ on a curve is the differential code $C_\{\Omega \}(D,G)$ . We study the parameters of such codes and state a lower bound for their minimum distance. Using this bound, one can study some examples of codes on surfaces, and in particular surfaces with Picard number $1$ like elliptic quadrics or some particular cubic surfaces. The parameters of some of the studied codes reach those of the best known codes up to now.},
affiliation = {INRIA Saclay, Projet Tanc École polytechnique Laboratoire d’informatique LIX, UMR 7161 91128 Palaiseau Cedex, France},
author = {Couvreur, Alain},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {algebraic geometry codes; algebraic surfaces; differentials},
language = {eng},
month = {3},
number = {1},
pages = {95-120},
publisher = {Société Arithmétique de Bordeaux},
title = {Differential approach for the study of duals of algebraic-geometric codes on surfaces},
url = {http://eudml.org/doc/219685},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Couvreur, Alain
TI - Differential approach for the study of duals of algebraic-geometric codes on surfaces
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/3//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 1
SP - 95
EP - 120
AB - The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this description is less trivial, it can be regarded as a natural extension to surfaces of the result asserting that the dual of a functional code $C_L (D,G)$ on a curve is the differential code $C_{\Omega }(D,G)$ . We study the parameters of such codes and state a lower bound for their minimum distance. Using this bound, one can study some examples of codes on surfaces, and in particular surfaces with Picard number $1$ like elliptic quadrics or some particular cubic surfaces. The parameters of some of the studied codes reach those of the best known codes up to now.
LA - eng
KW - algebraic geometry codes; algebraic surfaces; differentials
UR - http://eudml.org/doc/219685
ER -

References

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