On $r$-primitive $k$-normal polynomials with two prescribed coefficients

This article investigates the existence of an $r$-primitive $k$-normal polynomial, defined as the minimal polynomial of an $r$-primitive $k$-normal element in ${\mathbb{F}}_{q^n}$, with a specified degree $n$ and two given coefficients over the finite field ${\mathbb{F}}_q$. Here, $q$ represents an odd prime power, and $n$ is an integer. The article establishes a sufficient condition to ensure the existence of such a polynomial. Using this condition, it is demonstrated that a $2$-primitive $2$-normal polynomial of degree $n$ always exists over ${\mathbb{F}}_q$ when both $q\ge 11$ and $n\ge 15$. However, for the range $10\le n\le 14$, uncertainty remains regarding the existence of such a polynomial for $71$ specific pairs of $(q,n)$. Moreover, when $q<11$, the number of uncertain pairs reduces to $16$. Furthermore, for the case of $n=9$, extensive computational power is employed using SageMath software, and it is found that the count of such uncertain pairs is reduced to $3988$.