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

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.

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.

@article{JTNB_2011__23_1_95_0,
     author = {Alain Couvreur},
     title = {Differential approach for the study of duals of algebraic-geometric codes on surfaces},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {95--120},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {1},
     year = {2011},
     doi = {10.5802/jtnb.752},
     zbl = {1278.14036},
     mrnumber = {2780621},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.752/}
}
Alain Couvreur. Differential approach for the study of duals of algebraic-geometric codes on surfaces. Journal de Théorie des Nombres de Bordeaux, Tome 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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