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

Edoukou, Frédéric A. B.

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 131-143.

On étudie le code fonctionnel $C_2\left(X\right)$ défini sur une variété algébrique projective $X$, dans le cas où $X\subset {\mathbb{P}}^3\left({𝔽}_q\right)$ est une surface Hermitienne non-dégénérée. Nous donnons d’abord des bornes pour $\#X_{Z\left(𝒬\right)}\left({𝔽}_q\right)$ 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 $C_2\left(X\right)$.

We study the functional codes $C_2\left(X\right)$ defined on a projective algebraic variety $X$, in the case where $X\subset {\mathbb{P}}^3\left({𝔽}_q\right)$ is a non-degenerate Hermitian surface. We first give some bounds for $\#X_{Z\left(𝒬\right)}\left({𝔽}_q\right)$, 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 $C_2\left(X\right)$.

[1] Y. Aubry, M. Perret, On the characteristic polynomials of the Frobenuis endomorphism for projective curves over finite fields. Finite Fields and Theirs Applications 10 (2004), 412–431. | MR | Zbl

[2] R. C. Bose, I. M. Chakravarti, Hermitian varieties in finite projective space $PG(N,q)$. Canadian J. of Math. 18 (1966), 1161–1182. | MR | Zbl

[3] I. M. Chakravarti, The generalized Goppa codes and related discrete designs from Hermitian surfaces in $PG(3,s^2)$. Lecture Notes in Computer Science 311 (1986), 116–124. | MR | Zbl

[4] F. A. B. Edoukou, Codes defined by forms of degree 2 on Hermitian surface and Sørensen conjecture. Finite Fields and Their Applications, Volume 13, Issue 3 (2007), 616–627. | MR | Zbl

[5] R. Hartshorne, Algebraic Geometry. Graduate texts in mathematics 52, Springer-Verlag, 1977. | MR | Zbl

[6] J. W. P. Hirschfeld, Projective Geometries Over Finite Fields. (Second Edition) Clarendon Press. Oxford, 1998. | MR | Zbl

[7] J. W. P. Hirschfeld, Finite projective spaces of three dimensions. Clarendon press. Oxford, 1985. | MR | Zbl

[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. | MR | Zbl

[9] I. R. Shafarevich, Basic algebraic geometry 1. Springer-Verlag, 1994. | MR | Zbl

[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 $GF\left(4\right)$. Master’s Thesis, University of North Carolina at Chapell Hill, USA, 1986.