Consider the field $\mathbb{F}={\rm GF}(q=2^m)$ for $m \ge 2$. Denote by $\{\omega_1, \omega_2,...,\omega_m\}\in \mathbb{F}$ a basis to $\mathbb{F}$ over GF($2$). That is, each element in $\mathbb{F}$ can be written as a linear combination of the basis elements in $\{\omega_1, \omega_2,...,\omega_m\}$, where the coefficients are taken from GF($2$) (namely, the coefficients are $0$ or $1$).
I am trying to prove the existence of an element $g \in \mathbb{F}$ such that each of the $m-1$ sets $\{g\omega_1, \omega_1,\omega_2,...,\omega_{m-1}\}$, $\{g\omega_1, g\omega_2,\omega_1,\omega_2,...,\omega_{m-2}\}$ ... $\{g\omega_1, g\omega_2,...,g\omega_{m-1},\omega_1\}$ is a basis to $\mathbb{F}$ over GF($2$). In other words, $g$ must guarantee the linear independence of the elements in each of the sets above.
I managed to show experimentally (using a polynomial basis) that for each $m$ up to $12$ there are exactly two such $g$'s, where one is the multiplicative inverse of the other. However, I was not able so far to find any structure in the $g$'s. One simple observation I have is that $g$ must be of the form $w_m + x$ for some $x \ne w_m$ to satisfy the linear independence of $\{g\omega_1, \omega_1,\omega_2,...,\omega_{m-1}\}$.
Comment: It is fine as well to show the existence of such a $g$ for a particular choice of a basis.
Example
Assume that $m=4$ and that $\{\omega_1, \omega_2,\omega_3,\omega_4\}$ is a basis to ${\rm GF}(2^4 = 16)$ over $\rm{GF}(2)$. We need to find an element $g \in \rm{GF}(16)$ such that each of the sets $\{g\omega_1, \omega_1, \omega_2, \omega_3\}$, $\{g\omega_1, g\omega_2, \omega_1, \omega_2\}$, $\{g\omega_1, g\omega_2, g\omega_3, \omega_1\}$ is a basis to ${\rm GF}(16)$ over ${\rm GF}(2)$. Suppose we work with the polynomial basis $\{\omega_1 = 1, \omega_2 = \alpha,\omega_3 = \alpha^2,\omega_4 = \alpha^3\}$ where $\alpha$ is a root to the primitive polynomial $D^4 + D + 1 =0$. In this case, there are two possible $g$'s: $g_1 = \alpha^3 + \alpha$ and $g_2 = \alpha^3 + \alpha^2$, where $g_1 \cdot g_2 =1$.
Observation 1
It seems that it suffices to guarantee that $g \cdot \omega_i = \omega_{m-i+1} + x_i$ for $x_i \ne \omega_{m-i+1}$.
Observation 2
For odd $s$, there exists a construction of GF($2^s$) such that $g_1,g_2$ are the roots of the polynomial $D^2+D+1$ (which is irreducible over GF($2$)). For even $s$, there exists a construction of GF($2^s$) such that $g_1,g_2$ are the roots of the polynomial $D^2+(1+\alpha)D+1$, where $\alpha$ is the root of a primitive polynomial. This is true up to $s=12$.