The trace distribution of $r$-primitive $k$-normal elements of finite fields

Abstract

Let $q=p^e$ be a prime power and $n\geq1$. 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 $r\mid q^n-1$, $0\leq k\le n-1$, and $a\in K$, let $\mathcal{N}_{r,k}(a)$ denote the number of $r$-primitive $k$-normal elements of $L$ with trace $a$. We establish a sufficient condition for $\mathcal{N}_{r,k}$ to be constant on $K^\times$. We also identify a subgroup $H_r$ of $K^\times$ and prove that $\mathcal{N}_{r,k}$ is constant on each coset of $H_r$. Finally, we obtain a necessary and sufficient character-sum condition for $\mathcal{N}_{r,k}$ to be constant on $K^\times$.