Vanishing ideals over finite fields

TL;DR

This paper derives a formula for the vanishing ideal of a subset of projective space over a finite field, enabling computation of algebraic invariants and parameters of related Reed-Muller-type codes.

Contribution

It provides a new explicit formula for the vanishing ideal over finite fields, facilitating analysis of algebraic invariants and code parameters.

Findings

Formula for the vanishing ideal $I(X)$ over finite fields.

Method to compute algebraic invariants of $I(X)$.

Determination of basic parameters of Reed-Muller-type codes.

Abstract

Let $\mathbb{F}_q$ be a finite field, let $\mathbb{X}$ be a subset of a projective space ${\mathbb P}^{s-1}$, over the field $\mathbb{F}_q$, parameterized by rational functions, and let $I(\mathbb{X})$ be the vanishing ideal of $\mathbb{X}$. The main result of this paper is a formula for $I(\mathbb{X})$ that will allows us to compute: (i) the algebraic invariants of $I(\mathbb{X})$, and (ii) the basic parameters of the corresponding Reed-Muller-type code.