r-primitive k-normal elements in arithmetic progressions over finite fields

2022/11/03 by Josimar J. R. Aguirre, Aguirre, Josimar J. R., Abílio Lemos +5 · 1 citation
Computer Science · Social Sciences · #11T23 #12E20 #Coding theory and cryptography #FOS: Mathematics #Historical Geopolitical and Social Dynamics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2211.02114

openalex publication_date 2022/11/03 · openalex created_date 2023/02/15 · openalex updated_date 2026/07/28

Abstract

Let \mathbbF_qⁿ be a finite field with qⁿ elements. For a positive divisor r of qⁿ-1, the element α∈ \mathbbF_qⁿ^* is called r-primitive if its multiplicative order is (qⁿ-1)/r. Also, for a non-negative integer k, the element α∈ \mathbbF_qⁿ is k-normal over \mathbbF_q if gcd(αx^n-1+ α^q x^n-2 + … + α^q^n-2x + α^q^n-1 , xⁿ-1) in \mathbbF_qⁿ[x] has degree k. In this paper we discuss the existence of elements in arithmetic progressions \α, α+β, α+2β, …α+(m-1)β\ ⊂ \mathbbF_qⁿ with α+(i-1)β being r_i-primitive and at least one of the elements in the arithmetic progression being k-normal over \mathbbF_q. We obtain asymptotic results for general k, r₁, …, r_m and concrete results when k = r_i = 2 for i ∈ \1, …, m\.

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• On the Sum of Additive Characters and its Applications over Finite Fields

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