Weierstrass Semigroup, Pure Gaps and Codes on Function Fields
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911947325177856 |
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| author | Castellanos, Alonso S. Mendoza, Erik A. R. Quoos, Luciane |
| author_facet | Castellanos, Alonso S. Mendoza, Erik A. R. Quoos, Luciane |
| contents | We determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation $y^{m}=\prod_{i=1}^{r} (x-α_i)^{λ_i}$ over $K$, the algebraic closure of $\mathbb{F}_q$, where $α_1, \dots, α_r\in K$ are pairwise distinct elements, and $\gcd(m, \sum_{i=1}^{r}λ_i)=1$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_02128 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weierstrass Semigroup, Pure Gaps and Codes on Function Fields Castellanos, Alonso S. Mendoza, Erik A. R. Quoos, Luciane Algebraic Geometry 94B27, 11G20, 14H55 We determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation $y^{m}=\prod_{i=1}^{r} (x-α_i)^{λ_i}$ over $K$, the algebraic closure of $\mathbb{F}_q$, where $α_1, \dots, α_r\in K$ are pairwise distinct elements, and $\gcd(m, \sum_{i=1}^{r}λ_i)=1$. 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. |
| title | Weierstrass Semigroup, Pure Gaps and Codes on Function Fields |
| topic | Algebraic Geometry 94B27, 11G20, 14H55 |
| url | https://arxiv.org/abs/2304.02128 |