About r-primitive and k-normal elements in finite fields

Abstract

Let F q n be a finite field with q n elements and r be a positive divisor of q n − 1. An element α ∈ F * q n is called r-primitive if its multiplicative order is (q n −1)/r. Also, α ∈ F q n is k-normal over F q if the greatest common divisor of the polynomials g α (x) = αx n−1 +α q x n−2 +. . .+α q n−2 x+α q n−1 and x n −1 in F q n [x] has degree k. These concepts generalize the ideas of primitive and normal elements, respectively. In this paper, we consider non-negative integers m 1 , m 2 , k 1 , k 2 , positive integers r 1 , r 2 and rational functions F (x) = F 1 (x)/F 2 (x) ∈ F q n (x) with deg(F i ) ≤ m i for i ∈ {1, 2} satisfying certain conditions and we present sufficient conditions for the existence of r 1 -primitive k 1 -normal elements α ∈ F q n over F q , such that F (α) is an r 2 -primitive k 2 -normal element over F q . Finally as an example we study the case where r 1 = 2, r 2 = 3, k 1 = 2, k 2 = 1, m 1 = 2 and m 2 = 1, with n ≥ 7.

Seismic Performance of Vertically Prestressed Semicircular Anti-Slide Piles Based on Deep Machine-learning Algorithms

“…Currently, no relevant research results have been reported in the world, and the relevant scientific research process of this structural type is significantly behind the production and practical application, which is worthy of further research. Some scientists have used the full power finite element method to compare the dynamic response of slopes supported by prestressed anchor cables and unsupported slopes during earthquakes [1][2]. Some experts believe that trapezoidal anchored prestressed vertical anti-skid rods are a new method for treating water damage and landslides [3][4].…”