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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 Hermitianvariety. We give a condition of divisibility of the weights of the codewords.For $\mathcal{X}$ a non-degenerate Hermitian surface, we list the first fiveweights and the corresponding codewords and give a positive answer on aconjecture formulated on this question. The paper ends with a conjecture on theminimum 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 Hermitianvariety.

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