The weight distribution of the functional codes defined by forms of degree 2 on Hermitian surfaces
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, p. 131-143

We study the functional codes C 2 (X) defined on a projective algebraic variety X, in the case where X 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).

On étudie le code fonctionnel C 2 (X) défini sur une variété algébrique projective X, dans le cas où X 3 (𝔽 q ) est une surface Hermitienne non-dégénérée. Nous donnons d’abord des bornes pour #X Z(𝒬) (𝔽 q ) meilleures que celles connues. Ensuite nous calculons le nombre de mots de code atteignant le second poids. Nous donnons aussi une estimation exacte du troisième poids, une description de la structure géométrique des mots correspondant, ainsi que leur nombre. L’article s’achève par une conjecture formulée sur les quatrième et cinquiéme poids du code C 2 (X).

@article{JTNB_2009__21_1_131_0,
     author = {Edoukou, Fr\'ed\'eric A. B.},
     title = {The weight distribution of the functional codes defined by forms of degree 2 on Hermitian surfaces},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {1},
     year = {2009},
     pages = {131-143},
     doi = {10.5802/jtnb.662},
     mrnumber = {2537708},
     zbl = {1183.94060},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_1_131_0}
}
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, Volume 21 (2009) no. 1, pp. 131-143. doi : 10.5802/jtnb.662. http://www.numdam.org/item/JTNB_2009__21_1_131_0/

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