On elements of high order in general finite fields

We show that the Gao's construction gives for any finite field $F_{q^{n}}$ elements with the multiplicative order at least $\binom{n+t-1}{t}\prod _{i=0}^{t-1}\frac{1}{d^{i}}$, where $d=\left\lceil 2\log _{q} n\right\rceil$, $t=\left\lfloor \log _{d} n\right\rfloor$.