  We study the dual linear code of points and generators on a non-singular
Hermitian variety $\mathcal{H}(2n+1,q^2)$. We improve the earlier results for
$n=2$, we solve the minimum distance problem for general $n$, we classify the
$n$ smallest types of code words and we characterize the small weight code
words as being a linear combination of these $n$ types.
