Nagy Gabor: Hermite-féle algebrai hibajavító kódok
Matthews and Michel investigated the minimum distances of 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
$\mathbb{F}_{q^2}(\mathcal{H})$. We determine the Weierstrass gap sequence
$G(P)$ where $P$ is a degree $3$ place of $\mathbb{F}_{q^2}(\mathcal{H})$,
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 to
Xing and Chen.