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 \(K^\times\).