Characterization of non-special divisors of small degree on Kummer extensions and LCP codes
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| Format: | Preprint |
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2026
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| _version_ | 1866911633535664128 |
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| author | Mendoza, Erik Navarro, Horacio Quoos, Luciane |
| author_facet | Mendoza, Erik Navarro, Horacio Quoos, Luciane |
| contents | A recent construction of linear complementary pairs (LCPs) of algebraic geometry codes is intimately linked to the identification of non-special divisors of small degree within a function field over a finite field. Let $\mathbb{F}_q$ be the finite field of cardinality $q$. In this work, we consider a function field $F/\mathbb{F}_q$ of genus $g$ defined by a Kummer extension of type $y^m = f(x)$, where $f(x)$ is a polynomial in $\mathbb{F}_q[x]$. Based on the theory of generalized Weierstrass semigroups at several places, we provide an arithmetic criterion to identify all non-special divisors of degree $g-1$ and $g$ whose support is contained in a subset of the totally ramified places of the extension $F/\mathbb{F}_q(x)$. Furthermore, we explicitly determine all non-special divisors of degree $g-1$ in certain cases. Finally, we apply these results to provide explicit new families of LCPs algebraic geometry codes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_27146 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Characterization of non-special divisors of small degree on Kummer extensions and LCP codes Mendoza, Erik Navarro, Horacio Quoos, Luciane Algebraic Geometry 11T71, 14G50, 14H55 A recent construction of linear complementary pairs (LCPs) of algebraic geometry codes is intimately linked to the identification of non-special divisors of small degree within a function field over a finite field. Let $\mathbb{F}_q$ be the finite field of cardinality $q$. In this work, we consider a function field $F/\mathbb{F}_q$ of genus $g$ defined by a Kummer extension of type $y^m = f(x)$, where $f(x)$ is a polynomial in $\mathbb{F}_q[x]$. Based on the theory of generalized Weierstrass semigroups at several places, we provide an arithmetic criterion to identify all non-special divisors of degree $g-1$ and $g$ whose support is contained in a subset of the totally ramified places of the extension $F/\mathbb{F}_q(x)$. Furthermore, we explicitly determine all non-special divisors of degree $g-1$ in certain cases. Finally, we apply these results to provide explicit new families of LCPs algebraic geometry codes. |
| title | Characterization of non-special divisors of small degree on Kummer extensions and LCP codes |
| topic | Algebraic Geometry 11T71, 14G50, 14H55 |
| url | https://arxiv.org/abs/2604.27146 |