arXiv subjects

Frederic A. B. Edoukou

Publications and source records attributed to Frederic A. B. Edoukou.

4 recordsLinked to original sources

arXiv2006

Codes defined by forms of degree 2 on quadric and non-degenerate hermitian varieties in $\mathbb{P}^{4}(\mathbb{F}_q)$}

We study the functional codes of second order defined by G. Lachaud on $\mathcal{X} \subset {\mathbb{P}}^4(\mathbb{F}_q)$ a quadric of rank($\mathcal{X}$)=3,4,5 or a non-degenerate hermitian variety. We give some bounds for %$# \mathcal{X}_{Z(\mathcal{Q})}(\mathbb{F}_{q})$, the number of points of quadratic sections of $\mathcal{X}$, which are the best possible and show that codes defined on non-degenerate quadrics are better than those defined on degenerate quadrics. We also show the geometric structure of the minimum weight codewords and estimate the second weight of these codes. For $\mathcal{X}$ a non-degenerate hermitian variety, we list the first five weights and the corresponding codewords. The paper ends with two conjectures. One on the minimum distance for the functional codes of order $h$ on $\mathcal{X} \subset {\mathbb{P}}^4(\mathbb{F}_q)$ a non-singular hermitian variety. The second conjecture on the distribution of the codewords of the first five weights of the functional codes of second order on $\mathcal{X} \subset {\mathbb{P}}^N(\mathbb{F}_q)$ the non-singular hermitian variety.

math.AG↗

arXiv2006

Codes defined by forms of degree 2 on hermitian surfaces and Sørensen's conjecture

We study the functional codes $C_h(X)$ defined by G. Lachaud in $\lbrack 10 \rbrack$ where $X \subset {\mathbb{P}}^N$ is an algebraic projective variety of degree $d$ and dimension $m$. When $X$ is a hermitian surface in $PG(3,q)$, Sørensen in \lbrack 15\rbrack, has conjectured for $h\le t$ (where $q=t^2$) the following result : $$# X_{Z(f)}(\mathbb{F}_{q}) \le h(t^{3}+ t^{2}-t)+t+1$$ which should give the exact value of the minimum distance of the functional code $C_h(X)$. In this paper we resolve the conjecture of Sørensen in the case of quadrics (i.e. $h=2$), we show the geometrical structure of the minimum weight codewords and their number; we also estimate the second weight and the geometrical structure of the codewords reaching this second weight

math.AG↗

arXiv2005

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

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

math.AG↗

arXiv2005

Codes Defined By Forms Of Degree 2 On Quadric Surfaces

We study the functional codes $C_2(X)$ defined on projective varieties $X$, in the case where $X\subset \mathbb{P}^3$ is a 1-degenerate quadric or a non-degenerate quadric (hyperbolic or elliptic). We find the minimum distance of these codes, the second weight, and the third weight. We also show the geometrical structure of the first weight and second weight codewords. One result states that the codes $C_2(X)$ defined on the elliptic quadrics are good codes according to the table of A. E. Brouwer.

math.AG↗