Computing Vanishing Ideals for Toric Codes

TL;DR

This paper develops methods to compute vanishing ideals on toric varieties over finite fields, with applications to error-correcting codes, especially for weighted projective spaces and numerical semigroups.

Contribution

It introduces algorithms for computing generating sets of vanishing ideals on toric varieties, extending to weighted projective spaces and numerical semigroup rings.

Findings

Algorithms for computing vanishing ideals on toric varieties.

Explicit generators for ideals in weighted projective spaces.

Connections between vanishing ideals and numerical semigroup rings.

Abstract

Motivated by applications to the theory of error-correcting codes, we give methods for computing a generating set for the ideal generated by $\beta$-graded polynomials vanishing on certain subsets of a simplicial complete toric variety $X$ over a finite field $\mathbb{F}_q$, where $\beta$ is a $d\times r$ matrix whose columns generate a subsemigroup $\mathbb{N}\beta$ 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 $\beta=[w_1 \cdots w_r]$ is a row matrix corresponding to a numerical semigroup $\mathbb{N}\beta=\langle w_1,\dots,w_r \rangle$, $X$ 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 $\{w_1,\dots,w_r\}$.