Effective existence of primitive k-normal elements via affine spaces

Abstract

Let ${\mathbb{F}}_{q^n}$ be the finite field with $q^n$ elements. An element $\alpha \in {\mathbb{F}}_{q^n}$ is $k$-normal over ${\mathbb{F}}_q$ if $\alpha$ and its conjugates span an ${\mathbb{F}}_q$-subspace of dimension $n-k$. It is known that, for $q$ sufficiently large, a primitive $k$-normal element exists if and only if there exists a $k$-normal element free of binomials, but the lower bound on $q$ is not explicit. We count the $k$-normal elements in specific affine subspaces of ${\mathbb{F}}_{q^n}$ and show that, in some of them, all but at most one element is $k$-normal. Combining this with character sum estimates over affine spaces and a sieve, we obtain an explicit lower bound on $q$ above which a primitive $k$-normal element exists, provided there is a $k$-normal element free of binomials whose ${\mathbb{F}}_q$-order has a linear factor or is irreducible. As an application, we compute these bounds for $5\le n\le 8$ and give a complete characterization of the existence of primitive $k$-normal elements in ${\mathbb{F}}_{q^6}$ over ${\mathbb{F}}_q$.

19 pages