7 research outputs found

    Complete set of Pure Gaps in Function Fields

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    In this work, we provide a way to completely determine the set of pure gaps G0(P1,P2)G_0(P_1, P_2) at two rational places P1,P2P_1, P_2 in a function field FF over a finite field Fq\mathbb{F}_q, and its cardinality. Furthermore, we given a bound for the cardinality of the set G0(P1,P2)G_0(P_1, P_2) which is better, in some cases, than the generic bound given by Homma and Kim. As a consequence, we completely determine the set of pure gaps and its cardinality for two families of function fields: the GKGK function field and Kummer extensions.Comment: 22 page

    Weierstrass Semigroup, Pure Gaps and Codes on Kummer Extensions

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    We determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation ym=∏i=1r(xβˆ’Ξ±i)Ξ»iy^{m}=\prod_{i=1}^{r} (x-\alpha_i)^{\lambda_i} over KK, the algebraic closure of Fq\mathbb{F}_q, where Ξ±1,…,Ξ±r∈K\alpha_1, \dots, \alpha_r\in K are pairwise distinct elements, and gcd⁑(m,βˆ‘i=1rΞ»i)=1\gcd(m, \sum_{i=1}^{r}\lambda_i)=1. For an arbitrary function field, from the knowledge of the minimal generating set of the Weierstrass semigroup at two rational places, the set of pure gaps is characterized. We apply these results to construct algebraic geometry codes over certain function fields with many rational places.Comment: 24 page

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

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    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 FF 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
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