تقرير
On $r$-primitive $k$-normal polynomials with two prescribed coefficients
| العنوان: | On $r$-primitive $k$-normal polynomials with two prescribed coefficients |
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| المؤلفون: | Sharma, Avnish K., Rani, Mamta, Tiwari, Sharwan K., Panigrahi, Anupama |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Number Theory, 12E20, 11T23 |
| الوصف: | 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 $3988$. Comment: 27 pages, 3 Tables |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.20760 |
| رقم الأكسشن: | edsarx.2405.20760 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |