Computing Vanishing Ideals for Toric Codes
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910974162763776 |
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| author | Şahin, Mesut |
| author_facet | Şahin, Mesut |
| contents | Motivated by applications to the theory of error-correcting codes, we give methods for computing a generating set for the ideal generated by $β$-graded polynomials vanishing on certain subsets of a simplicial complete toric variety $X$ over a finite field $\mathbb{F}_q$, where $β$ is a $d\times r$ matrix whose columns generate a subsemigroup $\mathbb{N}β$ of $\mathbb{N}^d$. We also give a method for computing the vanishing ideal of the set of $\mathbb{F}_q$-rational points of $X$. When $β=[w_1 \cdots w_r]$ is a row matrix corresponding to a numerical semigroup $\mathbb{N}β=\langle w_1,\dots,w_r \rangle$, $X$ is a weighted projective space and generators of the relevant vanishing ideal is given using generators of defining (toric) ideals of numerical semigroup rings corresponding to semigroups generated by subsets of $\{w_1,\dots,w_r\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_01061 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Computing Vanishing Ideals for Toric Codes Şahin, Mesut Algebraic Geometry Information Theory Commutative Algebra Combinatorics Primary 14M25, 14G05, Secondary 94B27, 11T71 Motivated by applications to the theory of error-correcting codes, we give methods for computing a generating set for the ideal generated by $β$-graded polynomials vanishing on certain subsets of a simplicial complete toric variety $X$ over a finite field $\mathbb{F}_q$, where $β$ is a $d\times r$ matrix whose columns generate a subsemigroup $\mathbb{N}β$ of $\mathbb{N}^d$. We also give a method for computing the vanishing ideal of the set of $\mathbb{F}_q$-rational points of $X$. When $β=[w_1 \cdots w_r]$ is a row matrix corresponding to a numerical semigroup $\mathbb{N}β=\langle w_1,\dots,w_r \rangle$, $X$ is a weighted projective space and generators of the relevant vanishing ideal is given using generators of defining (toric) ideals of numerical semigroup rings corresponding to semigroups generated by subsets of $\{w_1,\dots,w_r\}$. |
| title | Computing Vanishing Ideals for Toric Codes |
| topic | Algebraic Geometry Information Theory Commutative Algebra Combinatorics Primary 14M25, 14G05, Secondary 94B27, 11T71 |
| url | https://arxiv.org/abs/2207.01061 |