Consider a generalized Vandermonde matrix for $n$ vectors $(v_{1,1}, v_{1,2}, \cdots, v_{1,m}), (v_{2,1}, v_{2,2}, \cdots, v_{2,m}), \cdots, (v_{n,1}, v_{n,2},\cdots, v_{n,m})$, each of which is $\in \mathbb{Z}_q^m$, for some $m<n, q>n$: \begin{bmatrix}v_{1,1}&v_{1,1}^2 & \cdots&v_{1,1}^n & v_{1,2}&v_{1,2}^2 & \cdots&v_{1,2}^n &\cdots v_{1,n}&v_{1,n}^2 & \cdots&v_{1,m}^n \\ v_{2,1}&v_{2,1}^2 & \cdots&v_{2,1}^n & v_{2,2}&v_{2,2}^2 & \cdots&v_{2,2}^n &\cdots v_{2,n}&v_{2,n}^2 & \cdots&v_{2,m}^n \\ &&&&&&\vdots\\ v_{n,1}&v_{n,1}^2 & \cdots&v_{n,1}^n & v_{n,2}&v_{n,2}^2 & \cdots&v_{n,2}^n &\cdots v_{n,n}&v_{n,n}^2 & \cdots&v_{n,m}^n \end{bmatrix}
The sufficient requirement for this to be linearly independent is for some $i\in[m]$, such that $\{v_{j,i}\}_{j\in[n]}$ are all different (i.e., one of the column has all different elements). I wonder what is the sufficient and necessary (i.e., the minimal) requirement for a generalized vandermonde matrix to have full rank $n$?