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

TL;DR

This paper studies the existence of special polynomials over finite fields with prescribed coefficients, providing conditions for their existence and computational results for specific cases.

Contribution

It establishes a sufficient condition for the existence of $r$-primitive $k$-normal polynomials with two prescribed coefficients and applies it to various cases, including computational verification.

Findings

A $2$-primitive $2$-normal polynomial of degree $n$ exists for $q extgreater=11$ and $n extgreater=15$.

Uncertain cases remain for certain pairs $(q,n)$ with $10 extless= n extless=14$ and $q extless 11$.

Computational analysis reduces the number of uncertain pairs for $n=9$ to 3988.

Abstract

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 $q\geq 11$ and $n\geq 15$. However, for the range $10\leq n\leq 14$, 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 $3988$.