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Weierstrass semigroups, pure gaps and codes on function fields

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Abstract

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. Furthermore, we determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation \(y^{m}=\prod _{i=1}^{r} (x-\alpha _i)^{\lambda _i}\) over K, the algebraic closure of \({\mathbb {F}}_q\), where \(\alpha _1, \dots , \alpha _r\in K\) are pairwise distinct elements, \(1\le \lambda _i < m\), and \(\gcd (m, \sum _{i=1}^{r}\lambda _i)=1\). We apply these results to construct algebraic geometry codes over certain function fields with many rational places. For one-point codes we obtain families of codes with exact parameters.

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Authors and Affiliations

  1. Faculdade de Matemática, Universidade Federal de Uberlândia, Campus Santa Mânica, Uberlândia, CEP 38400-902, Brazil

    Alonso S. Castellanos

  2. IMECC, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda 651, Campinas, CEP 13083-859, Brazil

    Erik A. R. Mendoza

  3. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária, Rio de Janeiro, CEP 21941-909, Brazil

    Luciane Quoos

Authors
  1. Alonso S. Castellanos
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  2. Erik A. R. Mendoza
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  3. Luciane Quoos
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Corresponding author

Correspondence to Alonso S. Castellanos.

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Communicated by G. Lunardon.

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Alonso S. Castellanos was partially supported by FAPEMIG: APQ 00696-18 and RED 0013-21. Erik A. R. Mendoza was partially supported by FAPERJ Grant 201.650/2021 and FAPESP Grant 2022/16369-2. Luciane Quoos thanks FAPERJ 260003/001703/2021 - APQ1, CNPQ PQ 302727/2019-1 and CAPES MATH AMSUD 88881.647739/2021-01 for the partial support.

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Castellanos, A.S., Mendoza, E.A.R. & Quoos, L. Weierstrass semigroups, pure gaps and codes on function fields. Des. Codes Cryptogr. (2023). https://doi.org/10.1007/s10623-023-01339-w

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  • DOI: https://doi.org/10.1007/s10623-023-01339-w

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