Number of k-normal elements over a finite field

2022/02/20 by Aguirre, Josimar J. R., Neumann, Victor G. L.
#11T30 #12E20 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2202.09866

Abstract

An element α∈ \mathbbF_qⁿ is a normal element over \mathbbF_q if the conjugates α^qi, 0 ≤ i ≤ n-1, are linearly independent over \mathbbF_q. Hence a normal basis for \mathbbF_qⁿ over \mathbbF_q is of the form \α,α^q, …, α^q^n-1\, where α∈ \mathbbF_qⁿ is normal over \mathbbF_q. In 2013, Huczynska, Mullen, Panario and Thomson introduce the concept of k-normal elements, as a generalization of the notion of normal elements. In the last few years, several results have been known about these numbers. In this paper, we give an explicit combinatorial formula for the number of k-normal elements in the general case, answering an open problem proposed by Huczynska et al. (2013).

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