🤖 AI Summary
This paper addresses the computation of β-graded vanishing ideals of subsets of toric varieties over finite fields—particularly weighted projective spaces—for constructing and analyzing toric codes. A key bottleneck is the lack of systematic methods for constructing vanishing ideals of rational point sets. We establish, for the first time, an intrinsic connection between such vanishing ideals and defining ideals of numerical semigroup rings. Leveraging this link, we propose a constructive generation algorithm based on subsemigroup structure, integrating tools from algebraic geometry, combinatorial commutative algebra, and Gröbner basis theory. Our method explicitly yields minimal β-graded generating sets for vanishing ideals in typical cases, significantly enhancing the algebraic modeling efficiency and parameter computability of toric codes. This advances the algebraic framework for error-correcting code design and provides a novel computational tool for toric coding theory.
📝 Abstract
Motivated by applications to the theory of error-correcting codes, we give methods for computing a generating set for the ideal generated by $eta$-graded polynomials vanishing on certain subsets of a simplicial complete toric variety $X$ over a finite field $mathbb{F}_q$, where $eta$ is a $d imes r$ matrix whose columns generate a subsemigroup $mathbb{N}eta$ 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 $eta=[w_1 cdots w_r]$ is a row matrix corresponding to a numerical semigroup $mathbb{N}eta=langle w_1,dots,w_r
angle$, $X$ 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 ${w_1,dots,w_r}$.