On the vanishing ideal of an algebraic toric set and its parameterized linear codes

TL;DR

This paper investigates the algebraic structure of vanishing ideals of algebraic toric sets parameterized by clutter edges over finite fields, providing bounds, classifications, and estimates relevant to coding theory.

Contribution

It offers new estimates for the degree-complexity of vanishing ideals, classifies when these ideals are complete intersections for uniform clutters, and bounds the minimum distance of related linear codes.

Findings

Degree-complexity bounds for vanishing ideals

Classification of complete intersection property for uniform clutters

Upper bounds for minimum distance of parameterized linear codes

Abstract

Let K be a finite field and let X be a subset of a projective space, over the field K, which is parameterized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex order, of the vanishing ideal I(X) of X. If the clutter is uniform, we classify the complete intersection property of I(X) using linear algebra. We show an upper bound for the minimum distance of certain parameterized linear codes along with certain estimates for the algebraic invariants of I(X).