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

TL;DR

This work investigates the existence of $r$-primitive $k$-normal polynomials over finite fields with the last two coefficients prescribed, reframing the problem in terms of $r$-primitive $k$-normal elements whose trace and norm meet given values. A general framework is developed, yielding a concrete sufficiency condition $q^{\frac{m}{2}-k-2}> r\,W(Q)\,W\left(\frac{x^m-1}{g}\right)$ for the existence of such an element, with a refined prime-sieve approach improving the bound. The authors specialize to the case $(r,k)=(3,1)$ and determine, for $m\ge 7$, the exceptional $(q,m)$ pairs for which a $3$-primitive $1$-normal polynomial with prescribed last-two coefficients may not exist, while providing a comprehensive treatment of small $q$ and $m$. These results extend Hansen–Mullen–type coefficient-prescribing investigations to the broader class of $r$-primitive $k$-normal polynomials, offering practical criteria for constructive applications in finite fields.

Abstract

Let $ξ\in\mathbb{F}_{q^m}$ 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 $ξ$ is referred to be the $r$-primitive $k$-normal polynomial of $ξ$ 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 $m\geq 7$.

Theorems & Definitions (37)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Definition 2.5
• Lemma 2.6
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4