Suzuki-invariant codes from the Suzuki curve

In this paper we consider the Suzuki curve $y^q + y = x^{q_0}(x^q + x)$ over the field with $q = 2^{2m+1}$ elements. The automorphism group of this curve is known to be the Suzuki group $Sz(q)$ with $q^2(q-1)(q^2+1)$ elements. We construct AG codes over $\mathbb{F}_{q^4}$ from a $Sz(q)$-invariant divisor $D$, giving an explicit basis for the Riemann-Roch space $L(\ell D)$ for $0 < \ell \leq q^2-1$. These codes then have the full Suzuki group $Sz(q)$ as their automorphism group. These families of codes have very good parameters and are explicitly constructed with information rate close to one. The dual codes of these families are of the same kind if $2g-1 \leq \ell \leq q^2-1$

Generalized Weierstrass semigroups and their Poincaré series

Producción CientíficaWe investigate the structure of the generalized Weierstraß semigroups at several points on a curve defined over a finite field. We present a description of these semigroups that enables us to deduce properties concerned with the arithmetical structure of divisors supported on the specified points and their corresponding Riemann-Roch spaces. This characterization allows us to show that the Poincar´e series associated with generalized Weierstraß semigroups carry essential information to describe entirely their respective semigroups.Ministerio de Economía, Industria y Competitividad ( grant MTM2015-65764-C3-2-P / MTM2016-81735-REDT / MTM2016-81932-REDT)Universitat Jaume I (grant P1-1B2015-02 / UJI-B2018-10)Consejo Nacional de Desarrollo Científico y Tecnológico (grants 201584/2015-8 / 159852/2014-5 / 310623/2017-0)IMAC-Institut de Matemàtiques i Aplicacions de Castell

The Set of Pure Gaps at Several Rational Places in Function Fields

In this work, using maximal elements in generalized Weierstrass semigroups and its relationship with pure gaps, we extend the results in \cite{CMT2024} and provide a way to completely determine the set of pure gaps at several rational places in an arbitrary function field $F$ over a finite field and its cardinality. As an example, we determine the cardinality and a simple explicit description of the set of pure gaps at several rational places distinct to the infinity place on Kummer extensions, which is a different characterization from that presented by Hu and Yang in \cite{HY2018}. Furthermore, we present some applications in coding theory and AG codes with good parameters