Generalized minimum distance functions

Using commutative algebra methods we study the generalized minimum distance function (gmd function) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. The number of solutions that a system of homogeneous polynomials has in any given finite set o...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2018-10
Main Authors: Gonzalez-Sarabia, Manuel, Martínez-Bernal, Jose, Villarreal, Rafael H, Vivares, Carlos E
Format: Article
Language:English
Subjects:
Codes
Combinatorial analysis
Fields (mathematics)
Footprints
Lower bounds
Mathematical analysis
Polynomials
Rings (mathematics)
Weight
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Using commutative algebra methods we study the generalized minimum distance function (gmd function) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. The number of solutions that a system of homogeneous polynomials has in any given finite set of projective points is expressed as the degree of a graded ideal. If \(\mathbb{X}\) is a set of projective points over a finite field and \(I\) is its vanishing ideal, we show that the gmd function and the Vasconcelos function of \(I\) are equal to the \(r\)-th generalized Hamming weight of the corresponding Reed-Muller-type code \(C_\mathbb{X}(d)\) of degree \(d\). We show that the generalized footprint function of \(I\) is a lower bound for the \(r\)-th generalized Hamming weight of \(C_\mathbb{X}(d)\). Then we present some applications to projective nested cartesian codes. To give applications of our lower bound to algebraic coding theory, we show an interesting integer inequality. Then we show an explicit formula and a combinatorial formula for the second generalized Hamming weight of an affine cartesian code.
ISSN:2331-8422