Recursive constructions of k-normal polynomials over finite fields

TL;DR

This paper introduces a recursive method to generate infinite sequences of k-normal polynomials over finite fields, which are useful in coding theory and cryptography, by applying a specific polynomial transformation iteratively.

Contribution

It presents a novel recursive construction technique for k-normal polynomials over finite fields using a specific transformation, expanding the toolkit for finite field polynomial generation.

Findings

Constructs infinite sequences of k-normal polynomials

Uses a specific transformation to generate polynomials

Applicable for various degrees and parameters

Abstract

The paper is devoted to produce infinite sequences of $k$-normal polynomials $F_{u}(x)\in \mathbb{F}_{q}[x]$ of degrees $np^{u} ~ (u\geq 0)$, for a suitably chosen initial $k$-normal polynomial $F_{0}(x)\in \mathbb{F}_{q}[x]$ of degree $n$ over $\mathbb{F}_{q}$ by iteratively applying the transformation $x\rightarrow \frac{x^p-x}{x^p-x+\delta}$, where $\delta\in \mathbb{F}_{q}$ and $0\leq k\leq n-1$.