On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^2)$

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.