  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 $C_2(X)$.
