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Projective nested cartesian codes

Cicero Carvalho, V. G. Lopez Neumann, Hiram H. Lopez

TL;DR

An upper bound for the minimum distance and the exact minimum distance in a special case (which includes the projective Reed–Muller codes) is calculated and some relations between the parameters of these codes and those of the affine cartesian codes are shown.

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.

Projective nested cartesian codes

TL;DR

An upper bound for the minimum distance and the exact minimum distance in a special case (which includes the projective Reed–Muller codes) is calculated and some relations between the parameters of these codes and those of the affine cartesian codes are shown.

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 , 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.

Paper Structure

This paper contains 3 sections, 20 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Length and dimension
  3. Minimum Distance

Key Result

Proposition 1.3

Let $I \subset S$ be an ideal and let $\{g_1, \ldots, g_s\}$ be a Gröbner basis for $I$. Then a monomial $M$ is in $\Delta(I)$ if and only if $M$ is not a multiple of $lm(g_i)$ for all $i = 1, \ldots, s$.

Theorems & Definitions (46)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • proof
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 1.8
  • ...and 36 more