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Evaluation of iterated Ore polynomials and skew Reed-Muller codes

Andre Leroy, Nabil Bennenni

Abstract

In this paper, we study two ways of evaluating iterated Ore polynomials. We provide many examples and compare these evaluations. We use the evaluation maps to construct Reed-Muller codes and compute explicitly some of the data that are associated to such codes.

Evaluation of iterated Ore polynomials and skew Reed-Muller codes

Abstract

In this paper, we study two ways of evaluating iterated Ore polynomials. We provide many examples and compare these evaluations. We use the evaluation maps to construct Reed-Muller codes and compute explicitly some of the data that are associated to such codes.
Paper Structure (4 sections, 5 theorems, 18 equations)

This paper contains 4 sections, 5 theorems, 18 equations.

Table of Contents

  1. Polynomial maps in $n$ variables.
  2. Skew Reed-Muller Codes Using Iterated Skew Polynomial Rings
  3. Iterated skew polynomial and Reed Muller codes
  4. Skew Reed Muller Codes Using Polynomial maps in $n$ variables.

Key Result

Theorem 1.6

Let $K$ be a ring and $R=K[t_1;\sigma_1,\delta_1][t_2;\sigma_2,\delta_2]\cdots[t_n;\sigma_n,\delta_n]$. We consider $(a_1,a_2,\dots,a_n)\in K^n$ and put $I=R(t-a_1)+R(t-a_2)+\dots + R(t-a_n)$ and $I_n=R_1(t_1-a_1)+ \dots +R_{n-1}(t_{n-1}-a_{n-1})+R(t_n-a_n)$, where, for $1\le i \le n$, $R_i$, $R_i=K

Theorems & Definitions (20)

  • Remark 1.1
  • Example 1.2
  • Definition 1.3
  • Remark 1.4
  • Example 1.5
  • Theorem 1.6
  • proof
  • Remark 1.7
  • Definition 1.8
  • Example 1.9
  • ...and 10 more