On the number of zero trace elements in polynomial bases for \(\mathbb F_{2^n}\) (Q558716)

Let \(\mathbb F_{2^n}\) denote the finite field of \(2^n\) elements, \(\alpha\) a defining element of \(\mathbb F_{2^n}\), i.e., \(\mathbb F_2(\alpha)=\mathbb F_{2^n}\), and denote by \(N(\alpha)\) the number of zero trace elements in the polynomial basis \(\{1,\alpha,\ldots,\alpha^{n-1}\}\). The author proves that there exists a defining element \(\alpha\in \mathbb F_{2^n}\) such that \[ N(\alpha)\geq n-\log n-2,\quad n\geq 21. \] The proof is based on the Weil bound. The question about the spectrum of possible values of \(N(\alpha)\), when \(\alpha\) runs through all defining elements of \(F_{2^n}\), has been introduced in \textit{O. Ahmadi} and \textit{A. Menezes} [On the number of trace-one elements in polynomial bases for \(\mathbb F_{2^n}\), Des. Codes Cryptography 37, No. 3, 493--507 (2005; Zbl 1142.11377)].