The Proportion of Weierstrass Semigroups

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 $\mathbb{F}_{q^4}\setminus\fq$-point whose parameters are better in some cases than the ones constructed in a similar way from an $\fq$-r…

超椭圆曲线循环覆盖的 Weierstrass 半群

Weierstrass semigroups from cyclic covers of hyperelliptic curves

本文研究了超椭圆曲线循环覆盖的 Weierstrass 半群可实现性问题，证明了其可实现性由超椭圆 Jacobi 簇中相关除子类在 $j$ 倍乘法下的行为决定。

Weierstrass sets on finite graphs

We study two possible tropical analogues of Weierstrass semigroups on graphs, called rank and functional Weierstrass sets. We prove that on simple graphs, the first is contained in the second. We completely characterize the subsets of N arising as a functional Weierstrass set of some graph. Finally, we give a sufficient condition for a subset of N to be the rank Weierstrass set of some graph, allowing us to construct examples of rank Weierstrass sets that are not semigroups.

数值半群的克利福德缺陷

The Clifford defect of a numerical semigroup

本文研究了来自曲线的数值半群的克利福德缺陷，并给出了其显式公式。

The second Feng-Rao number for codes coming from inductive semigroups

The second Feng-Rao number of every inductive numerical semigroup is explicitly computed. This number determines the asymptotical behaviour of the order bound for the second Hamming weight of one-point AG codes. In particular, this result is applied for the codes defined by asymptotically good towers of function fields whose Weierstrass semigroups are inductive. In addition, some properties of inductive numerical semigroups are studied, the involved Apéry sets are computed in a recursive way, and some tests to check whether a given numerical semigroups is inductive or not are provided.

The Buchweitz set of a numerical semigroup

Let $A \subset {\mathbb Z}$ be a finite subset. We denote by $\mathcal{B}(A)$ the set of all integers $n \ge 2$ such that $|nA| > (2n-1)(|A|-1)$, where $nA=A+\cdots+A$ denotes the $n$-fold sumset of $A$. The motivation to consider $\mathcal{B}(A)$ stems from Buchweitz's discovery in 1980 that if a numerical semigroup $S \subseteq {\mathbb N}$ is a Weierstrass semigroup, then $\mathcal{B}({\mathbb N} \setminus S) = \emptyset$. By constructing instances where this condition fails, Buchweitz disproved a longstanding conjecture by Hurwitz (1893). In this paper, we prove that for any numerical semi…