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The trace distribution of r-primitive k-normal elements of finite fields
Abstract
Let q=pe be a prime power and n≥1. Let L=Fqn and K=Fq. We study the trace distribution of r-primitive k-normal elements of L over K. We first derive exact formulas for the number of k-normal elements of L with a prescribed trace in K. For r| qn−1, 0≤ k≤ n−1, and a∈ K, let 𝒩r,k(a) denote the number of r-primitive k-normal elements of L with trace a. We establish a sufficient condition for 𝒩r,k to be constant on K×. We also identify a subgroup Hr of K× and prove that 𝒩r,k is constant on each coset of Hr. Finally, we obtain a necessary and sufficient character-sum condition for 𝒩r,k to be constant on K×.
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BibTeX
@misc{choudhary2026trace,
title = {{The trace distribution of $r$-primitive $k$-normal elements of finite fields}},
author = {Aakash Choudhary and Supriya Pisolkar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39510},
eprint = {2610.39510},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39510},
}
RIS
TY - UNPB
TI - The trace distribution of r-primitive k-normal elements of finite fields
AU - Aakash Choudhary
AU - Supriya Pisolkar
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39510
UR - https://arxiv.org/abs/2610.39510
AB - Let q=p^e be a prime power and n≥1. Let L=F_q^n and K=F_q. We study the trace distribution of r-primitive k-normal elements of L over K. We first derive exact formulas for the number of k-normal elements of L with a prescribed trace in K. For r| q^n−1, 0≤ k≤ n−1, and a∈ K, let 𝒩_r,k(a) denote the number of r-primitive k-normal elements of L with trace a. We establish a sufficient condition for 𝒩_r,k to be constant on K^×. We also identify a subgroup H_r of K^× and prove that 𝒩_r,k is constant on each coset of H_r. Finally, we obtain a necessary and sufficient character-sum condition for 𝒩_r,k to be constant on K^×.
ER -