Weierstrass semigroups at every point of the Suzuki curve

We explicitly determine the structure of the Weierstrass semigroups H(P) for any point P of the Suzuki curve q. As the point P varies, exactly two possibilities arise for H(P): one for the q-rational points (already known in the literature), and one for all remaining points. For this last case the minimal set of generators of H(P) is also provided. As an application, we construct dual one-point codes from an q4∖q-point whose parameters are better in some cases than the ones constructed in a similar way from an q-rational point.