Minimum distance functions of graded ideals and Reed-Muller-type codes

TL;DR

This paper introduces a new algebraic approach to analyze the minimum distance of Reed-Muller-type codes using graded ideals, Groebner bases, and Hilbert functions, providing bounds and supporting conjectures.

Contribution

It generalizes the minimum distance concept for projective Reed-Muller-type codes through algebraic invariants and offers a method to compute bounds using algebraic tools.

Findings

Established a generalized minimum distance function for graded ideals.

Provided a method to compute lower bounds for code minimum distances.

Presented explicit upper bounds for zeros of polynomials in projective sets.

Abstract

We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed-Muller-type codes over finite fields. This gives an algebraic formulation of the minimum distance of a projective Reed-Muller-type code in terms of the algebraic invariants and structure of the underlying vanishing ideal. Then we give a method, based on Groebner bases and Hilbert functions, to find lower bounds for the minimum distance of certain Reed-Muller-type codes. Finally we show explicit upper bounds for the number of zeros of polynomials in a projective nested cartesian set and give some support to a conjecture of Carvalho, Lopez-Neumann and Lopez.