Matthews and Michel investigated the minimum distances in certain
algebraic-geometry codes arising from a higher degree place P. In terms of
the Weierstrass gap sequence at P, they proved a bound that gives an
improvement on the designed minimum distance. In this paper, we consider those
of such codes which are constructed from the Hermitian function field. We
determine the Weierstrass gap sequence G(P) where P is a degree 3 place,
and compute the Matthews and Michel bound with the corresponding improvement.
We show more improvements using a different approach based on geometry. We also
compare our results with the true values of the minimum distances of Hermitian
1-point codes, as well as with estimates due Xing and Chen
