Weierstrass semigroups at every point of the Suzuki curve

In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Suzuki curve ${\mathcal{S}}_q$. As the point $P$ varies, exactly two possibilities arise for $H(P)$: one for the ${\mathbb{F}}_q$-rational points (already known in the literature), and one for all remaining points. For this last case a minimal set of generators of $H(P)$ is also provided. As an application, we construct dual one-point codes from an ${\mathbb{F}}_{q^4}\setminus \text{fq}$-point whose parameters are better in some cases than the ones constructed in a similar way from an $\text{fq}$-rational point.