Weierstrass semigroups at every point of the Suzuki curve
Volume 197 / 2021
Acta Arithmetica 197 (2021), 1-20
MSC: Primary 11G20; Secondary 11R58, 14H05, 14H55.
DOI: 10.4064/aa181203-24-2
Published online: 5 October 2020
Abstract
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 the 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 {\mathbb F_q} $-point whose parameters are better in some cases than the ones constructed in a similar way from an ${\mathbb F_q} $-rational point.
