Let H be the Hermitian curve defined over a finite field
Fq2β. In this paper we complete the geometrical characterization
of the supports of the minimum-weight codewords of the algebraic-geometry codes
over H, started in [1]: if d is the distance of the code, the
supports are all the sets of d distinct Fq2β-points on
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 dβ₯q2βq studied in [1], one of the two curves is always the Hermitian curve
H itself, while if d<q the supports are complete intersection of
two curves none of which can be H.
Finally, for some special codes among those with intermediate distance
between q and q2βq, 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)