A note on depth-$b$ normal elements

Abstract

In this paper we study elements $\beta \in {\mathbb{F}}_{q^n}$ having normal $\alpha$-depth $b$; that is, elements for which $\beta ,\beta -\alpha ,\dots ,\beta -(b-1)\alpha$ are simultaneously normal elements of ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$. In [1], the authors present the definition of normal $1$-depth but mistakenly present results for normal $\alpha$-depth for some fixed normal element $\alpha \in {\mathbb{F}}_{q^n}$. We explain this discrepancy and generalize the given definition of normal $(1-)$depth from [1] as well as answer some open questions presented in [1].