Assume that we have a $p \times p$ matrix $Z$ over $\mathbb{F}_{2^p}$
$$Z=\begin{bmatrix} w_1 & w_1^2& w_1^4& ... & {w_1}^{2^{\frac{p}{2}-1}} & \alpha_1w_1 & \alpha_1w_1^2& \alpha_1w_2^4& ... & \alpha_1{w_1}^{2^{\frac{p}{2}-1}}\\ w_2 & w_2^2&w_2^4& ... & {w_2}^{2^{\frac{p}{2}-1}} & \alpha_2w_2 & \alpha_2w_2^2&\alpha_2w_2^4& ... & \alpha_2{w_2}^{2^{\frac{p}{2}-1}}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots&\vdots& \vdots \\ w_p & w_p^2&w_p^4& ... & {w_p}^{2^{\frac{p}{2}-1}} & \alpha_pw_p & \alpha_pw_p^2& \alpha_pw_p^4& ... & \alpha_p{w_p}^{2^{\frac{p}{2}-1}}\\ \end{bmatrix}$$ where $w_i \in \mathbb{F}_{2^p}$ and $\alpha_i \in \mathbb{F}_{2^p}$ and $\forall \alpha_i$ is different from each other. How can I satisfy (which conditions are necessary) that this matrix has full rank?
P.S. This matrix construction is similar to the Moore matrix except half of the columns of $Z$ matrix is multiplied with $\alpha_i$s.
Thanks in advance.
Elif