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

TL;DR

This paper establishes duality criteria for projective Reed--Muller-type codes using v-numbers and Hilbert functions, extends duality theorems to Gorenstein ideals, and classifies self-dual codes via algebraic invariants.

Contribution

It introduces a global duality criterion based on v-numbers, extends duality results to Gorenstein vanishing ideals, and classifies self-dual codes using regularity and parity check matrices.

Findings

Provides a duality theorem for Gorenstein vanishing ideals

Classifies self-dual Reed-Muller-type codes over Gorenstein ideals

Shows how to compute the regularity index of generalized Hamming weights

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.