Projective nested cartesian codes

TL;DR

This paper introduces projective nested cartesian codes, a new class of codes derived from evaluating homogeneous polynomials on specific projective subsets, generalizing projective Reed-Muller codes and analyzing their parameters.

Contribution

It defines a new code family, computes their length and dimension, establishes bounds for minimum distance, and explores their relation to affine cartesian codes.

Findings

Calculated length and dimension of the codes.

Provided a lower bound and exact minimum distance in special cases.

Established relationships with affine cartesian codes.

Abstract

In this paper we introduce a new type of code, called projective nested cartesian code. It is obtained by the evaluation of homogeneous polynomials of a fixed degree on a certain subset of $\mathbb{P}^n(\mathbb{F}_q)$, and they may be seen as a generalization of the so-called projective Reed-Muller codes. We calculate the length and the dimension of such codes, a lower bound for the minimum distance and the exact minimum distance in a special case (which includes the projective Reed-Muller codes). At the end we show some relations between the parameters of these codes and those of the affine cartesian codes.