  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.
