Projective Parameterized Linear Codes Arising from some Matrices and their Main Parameters

TL;DR

This paper estimates key parameters of projective parameterized codes derived from matrices, providing formulas for length and dimension, and establishing bounds for minimum distance and regularity index, especially for codes related to graphs.

Contribution

It introduces formulas for code length and dimension using Hilbert functions and provides bounds for minimum distance and regularity index for specific classes of codes.

Findings

Derived formulas for code length and dimension.

Established upper bounds for minimum distance.

Provided lower bounds for regularity index in certain cases.

Abstract

In this paper we will estimate the main parameters of some evaluation codes which are known as projective parameterized codes. We will find the length of these codes and we will give a formula for the dimension in terms of the Hilbert function associated to two ideals, one of them being the vanishing ideal of the projective torus. Also we will find an upper bound for the minimum distance and, in some cases, we will give some lower bounds for the regularity index and the minimum distance. These lower bounds work in several cases, particularly for any projective parameterized code associated to the incidence matrix of uniform clutters and then they work in the case of graphs.