The trace spectra of polynomial bases for $${\mathbb{F}}_{2^{n}}$$

In this paper we study the trace spectra of polynomial bases for over . Shparlinski showed that there exists a polynomial basis having O(log n) elements of trace one. Here we show that for every t ≤ n, there exists a polynomial basis having t + O(log n) elements of trace one. We also study consequences of our results to the existence of irreducible polynomials of certain weights.