New informations on the structure of the functional codes defined by forms of degree $h$ on non-degenerate Hermitian varieties in $\mathbb{P}^{n(\mathbb{F}_q)$}

Abstract

We study the functional codes of order $h$ defined by G. Lachaud on $\mathcal{X}\subset {\mathbb{P}}^n({\mathbb{F}}_q)$ a non-degenerate Hermitian variety. We give a condition of divisibility of the weights of the codewords. For $\mathcal{X}$ a non-degenerate Hermitian surface, we list the first five weights and the corresponding codewords and give a positive answer on a conjecture formulated on this question. The paper ends with a conjecture on the minimum distance and the distribution of the codewords of the first $2h+1$ weights of the functional codes for the functional codes of order $h$ on $\mathcal{X}\subset {\mathbb{P}}^n({\mathbb{F}}_q)$ a non-singular Hermitian variety.

12 pages