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

Source: [$r$-primitive $k$-normal polynomials over finite fields with last two coefficients prescribed](<https://www.sota2.com/research/paper/68fa5d411e209131ce9d8877>)

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Data updated: 2025-10-25T18:09:08.151Z

Authors: K. Chatterjee, R.K. Sharma, S.K. Tiwari

Published: 2025-01-09T06:29:12.000Z

## Abstract

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

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2501.04999>)

[PDF](<https://arxiv.org/pdf/2501.04999.pdf>)