$r$-primitive $k$-normal polynomials over finite fields with last two coefficients prescribed

Abstract

Let $\hat{I}¾\in {\mathbb{F}}_{q^m}$ be an $r$-primitive $k$-normal element over ${\mathbb{F}}_q$, where $q$ is a prime power and $m$ is a positive integer. The minimal polynomial of $\hat{I}¾$ is referred to be the $r$-primitive $k$-normal polynomial of $\hat{I}¾$ over ${\mathbb{F}}_q$. In this article, we study the existence of an $r$-primitive $k$-normal polynomial over ${\mathbb{F}}_q$ such that the last two coefficients are prescribed. In this context, first, we prove a sufficient condition which guarantees the existence of such a polynomial. Further, we compute all possible exceptional pairs $(q,m)$ in case of $3$-primitive $1$-normal polynomials for $m\ge 7$.