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On $r$-primitive $k$-normal polynomials with two prescribed coefficients
- Publication Year :
- 2024
-
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 $3988$.<br />Comment: 27 pages, 3 Tables
- Subjects :
- Mathematics - Number Theory
12E20, 11T23
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2405.20760
- Document Type :
- Working Paper