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<p>In this article, we explore the properties of Weierstrass semigroups and pure gaps in the context of fiber products of Kummer extensions, with applications to algebraic geometry codes (AG codes). We establish an arithmetic criterion to determine gaps and pure gaps at totally ramified places in fiber products of Kummer extensions, and determine a system of generators for the Weierstrass semigroup at any totally ramified place, generalizing previous results for single Kummer extensions. Finally, we apply our results to specific maximal curves to construct AG codes with good relative parameters compared to existing codes derived from the GK curve.</p>

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On Weierstrass Semigroups and Fiber Product of Kummer Covers

  • 期刊论文

  • 发表日期：

    2025年11月6日

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  • Alonso S. Castellanos,
  • Erik Mendoza,
  • Guilherme Tizziotti

摘要

In this article, we explore the properties of Weierstrass semigroups and pure gaps in the context of fiber products of Kummer extensions, with applications to algebraic geometry codes (AG codes). We establish an arithmetic criterion to determine gaps and pure gaps at totally ramified places in fiber products of Kummer extensions, and determine a system of generators for the Weierstrass semigroup at any totally ramified place, generalizing previous results for single Kummer extensions. Finally, we apply our results to specific maximal curves to construct AG codes with good relative parameters compared to existing codes derived from the GK curve.