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

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\mi