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

Abstract

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.