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

Let \(q=p^e\) be a prime power and \(n≥1\). Let \(L=q^n\) and \(K=_q\). 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 \(Nr,k(a)\) 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 \(H_r\) of \(K^×\) and prove that \(Nr,k\) is constant on each coset of \(H_r\). Finally, we obtain a necessary and sufficient character-sum condition for \(Nr,k\) to be constant on \(K^×\).