  Let $\mathcal{H}$ be the Hermitian curve defined over a finite field
$\mathbb{F}_{q^2}$. In this paper we complete the geometrical characterization
of the supports of the minimum-weight codewords of the algebraic-geometry codes
over $\mathcal{H}$, started in [1]: if $d$ is the distance of the code, the
supports are all the sets of $d$ distinct $\mathbb{F}_{q^2}$-points on
$\mathcal{H}$ complete intersection of two curves defined by polynomials with
prescribed initial monomials w.r.t. \texttt{DegRevLex}.
  For most Hermitian codes, and especially for all those with distance $d\geq
q^2-q$ studied in [1], one of the two curves is always the Hermitian curve
$\mathcal{H}$ itself, while if $d<q$ the supports are complete intersection of
two curves none of which can be $\mathcal{H}$.
  Finally, for some special codes among those with intermediate distance
between $q$ and $q^2-q$, both possibilities occur.
  We provide simple and explicit numerical criteria that allow to decide for
each code what kind of supports its minimum-weight codewords have and to obtain
a parametric description of the family (or the two families) of the supports.
  [1] C. Marcolla and M. Roggero, Hermitian codes and complete intersections,
arXiv preprint arXiv:1510.03670 (2015).
