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Indicator functions, v-numbers and Gorenstein rings in the theory of projective Reed-Muller-type codes

Manuel González-Sarabia, Humberto Muñoz-George, Jorge A. Ordaz, Eduardo Sáenz-de-Cabezón, Rafael H. Villarreal

TL;DR

This paper develops a global duality framework for projective Reed-Muller-type codes using algebraic invariants of vanishing ideals. By linking the v-number ${\rm v}(I)$ and the Hilbert function $H_I(d)$ to the duality of projective codes, it extends known results from complete intersections to arbitrary Gorenstein vanishing ideals and provides a criterion that classifies self-dual codes via regularity and parity-check matrices. Central tools include indicator functions, essential monomials, and the $r$-th v-number ${\rm v}_r(I)$, which together describe generalized Hamming weights and the regularity index $R_r$ of the weight function. The results unify projective and affine duality theories, offer practical Macaulay2 procedures for computing duality data, and deepen connections between coding theory, Hilbert functions, and properties of Gorenstein vanishing ideals, with implications for constructing and analyzing dual codes from algebraic-geometry data.

Abstract

For projective Reed--Muller-type codes we give a global duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide a global duality theorem for projective Reed--Muller-type codes over Gorenstein vanishing ideals, generalizing the known case where the vanishing ideal is a complete intersection. We classify self dual Reed-Muller-type codes over Gorenstein ideals using the regularity and a parity check matrix. For projective evaluation codes, we give a duality theorem inspired by that of affine evaluation codes. We show how to compute the regularity index of the $r$-th generalized Hamming weight function in terms of the standard indicator functions of the set of evaluation points.

Indicator functions, v-numbers and Gorenstein rings in the theory of projective Reed-Muller-type codes

TL;DR

This paper develops a global duality framework for projective Reed-Muller-type codes using algebraic invariants of vanishing ideals. By linking the v-number and the Hilbert function to the duality of projective codes, it extends known results from complete intersections to arbitrary Gorenstein vanishing ideals and provides a criterion that classifies self-dual codes via regularity and parity-check matrices. Central tools include indicator functions, essential monomials, and the -th v-number , which together describe generalized Hamming weights and the regularity index of the weight function. The results unify projective and affine duality theories, offer practical Macaulay2 procedures for computing duality data, and deepen connections between coding theory, Hilbert functions, and properties of Gorenstein vanishing ideals, with implications for constructing and analyzing dual codes from algebraic-geometry data.

Abstract

For projective Reed--Muller-type codes we give a global duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide a global duality theorem for projective Reed--Muller-type codes over Gorenstein vanishing ideals, generalizing the known case where the vanishing ideal is a complete intersection. We classify self dual Reed-Muller-type codes over Gorenstein ideals using the regularity and a parity check matrix. For projective evaluation codes, we give a duality theorem inspired by that of affine evaluation codes. We show how to compute the regularity index of the -th generalized Hamming weight function in terms of the standard indicator functions of the set of evaluation points.
Paper Structure (12 sections, 35 theorems, 138 equations)

This paper contains 12 sections, 35 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: Commutative algebra and coding theory
  3. Commutative algebra
  4. Coding theory
  5. Linear codes
  6. Projective Reed--Muller-type codes
  7. Duality criteria for projective Reed--Muller-type codes
  8. Duality of projective evaluation codes and Gorenstein ideals
  9. Affine duality criterion
  10. The $r$-th v-number
  11. Examples
  12. Procedures for Macaulay$2$

Key Result

Lemma 2.1

hilbert-min-dis Let $\mathbb{X}$ be a finite subset of $\mathbb{P}^{s-1}$, let $[\alpha]$ be a point in $\mathbb{X}$ with $\alpha=(\alpha_1,\ldots,\alpha_s)$ and $\alpha_j\neq 0$ for some $j$, and let $I([\alpha])$ be the vanishing ideal of $[\alpha]$. Then $I([\alpha])$ is a prime ideal, ${\rm ht}(I([\alpha]))=s-1$, and $I(\mathbb{X})=\bigcap_{[\beta]\in{\mathbb{X}}}I([\beta])$ is the primary de

Theorems & Definitions (86)

  • Lemma 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Proposition 2.7
  • Lemma 2.8
  • proof
  • ...and 76 more