Differential approach for the study of duals of algebraic-geometric codes on surfaces
Journal de Théorie des Nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 95-120.

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.

L’objet de cet article est l’étude des orthogonaux de codes fonctionnels sur des surfaces algébriques. Nous en donnons une description géométrique directe à l’aide de formes différentielles. Bien que moins élémentaire, cette approche peut être vue comme une extension naturelle aux surfaces du résultat affirmant que l’orthogonal d’un code C L (D,G) sur une courbe est le code différentiel C Ω (D,G). Nous étudions les paramètres de ces codes et établissons un résultat de minoration de leur distance minimale. À l’aide de cette borne, on peut étudier certains exemples de codes sur des surfaces, en particulier sur des surfaces de nombre de Picard égal à 1 comme les quadriques elliptiques ou certaines surfaces cubiques. Les paramètres de certains codes étudiés égalent ceux des meilleurs codes connus à l’heure actuelle.

Published online:
DOI: 10.5802/jtnb.752
Alain Couvreur 1

1 INRIA Saclay, Projet Tanc École polytechnique Laboratoire d’informatique LIX, UMR 7161 91128 Palaiseau Cedex, France
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Alain Couvreur. Differential approach for the study of duals of algebraic-geometric codes on surfaces. Journal de Théorie des Nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 95-120. doi : 10.5802/jtnb.752. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.752/

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