Fix two $2^t$ length vector of form $p=\begin{bmatrix}u\_1&v\_1\end{bmatrix}\otimes\dots\otimes\begin{bmatrix}u\_t&v\_t\end{bmatrix}$ and $r=\begin{bmatrix}w\_1&z\_1\end{bmatrix}\otimes\dots\otimes\begin{bmatrix}w\_t&z\_t\end{bmatrix}$ where each $u\_i,v\_j,w\_{i'},z\_{j'}$ is a distinct prime.


Consider $2^r$ length vectors of form
$q(x\_1,y\_1,\dots,x\_r,y\_r)=\begin{bmatrix}x\_1&y\_1\end{bmatrix}\otimes\dots\otimes\begin{bmatrix}x\_r&v\_r\end{bmatrix}$ where $x\_i,y\_j\in\mathbb Z$ are allowed to vary.


Consider $2T\times 2^{r+t}$ matrices of form 
$$
\begin{bmatrix}p\otimes q(x\_1^{[1]},y\_1^{[1]},\dots,x\_r^{[1]},y\_r^{[1]})\\
p\otimes q(x\_1^{[2]},y\_1^{[2]},\dots,x\_r^{[2]},y\_r^{[2]})\\
p\otimes q(x\_1^{[3]},y\_1^{[3]},\dots,x\_r^{[3]},y\_r^{[3]})\\
\vdots\\
p\otimes q(x\_1^{[T-1]},y\_1^{[T-1]},\dots,x\_r^{[T-1]},y\_r^{[T-1]})\\
p\otimes q(x\_1^{[T]},y\_1^{[T]},\dots,x\_r^{[T]},y\_r^{[T]})\\
r\otimes q(x\_1^{[T+1]},y\_1^{[T+1]},\dots,x\_r^{[T+1]},y\_r^{[T+1]})\\
r\otimes q(x\_1^{[T+2]},y\_1^{[T+2]},\dots,x\_r^{[T+2]},y\_r^{[T+2]})\\
r\otimes q(x\_1^{[T+3]},y\_1^{[T+3]},\dots,x\_r^{[T+3]},y\_r^{[T+3]})\\
\vdots\\
r\otimes q(x\_1^{[2T-1]},y\_1^{[2T-1]},\dots,x\_r^{[2T-1]},y\_r^{[2T-1]})\\
r\otimes q(x\_1^{[2T]},y\_1^{[2T]},\dots,x\_r^{[2T]},y\_r^{[2T]})
\end{bmatrix}$$
and
$$
\begin{bmatrix}p\otimes q(x\_1^{[1]},y\_1^{[1]},\dots,x\_r^{[1]},y\_r^{[1]})\\
p\otimes q(x\_1^{[2]},y\_1^{[2]},\dots,x\_r^{[2]},y\_r^{[2]})\\
p\otimes q(x\_1^{[3]},y\_1^{[3]},\dots,x\_r^{[3]},y\_r^{[3]})\\
\vdots\\
p\otimes q(x\_1^{[T-1]},y\_1^{[T-1]},\dots,x\_r^{[T-1]},y\_r^{[T-1]})\\
p\otimes q(x\_1^{[T]},y\_1^{[T]},\dots,x\_r^{[T]},y\_r^{[T]})\\
q(x\_1^{[T+1]},y\_1^{[T+1]},\dots,x\_r^{[T+1]},y\_r^{[T+1]})\otimes r\\
q(x\_1^{[T+2]},y\_1^{[T+2]},\dots,x\_r^{[T+2]},y\_r^{[T+2]})\otimes r\\
q(x\_1^{[T+3]},y\_1^{[T+3]},\dots,x\_r^{[T+3]},y\_r^{[T+3]})\otimes r\\
\vdots\\
q(x\_1^{[2T-1]},y\_1^{[2T-1]},\dots,x\_r^{[2T-1]},y\_r^{[2T-1]})\otimes r\\
q(x\_1^{[2T]},y\_1^{[2T]},\dots,x\_r^{[2T]},y\_r^{[2T]})\otimes r
\end{bmatrix}$$
where $(x\_1^{[t]},y\_1^{[t},\dots,x\_r^{[t]},y\_r^{[t]})$ stands for $t$th choice of $x\_i,y\_j$ and $T$ is a positive integer.


How large can the rank of the matrices can be if $T=2^{r+t}$ holds?



On ranks of matrices with tensor structure
Clearly it can't exceed $2^r$ since each row is a linear function of the vector $q(...)$. To show that this upper bound is reached, we just have to show that we can take the $x,y$ such that 
$$
Q = 
\begin{bmatrix}
q^{[1]}\\ \vdots \\
q^{[2^r]}
\end{bmatrix}
$$
has full rank.


Take $[x\_i, y\_i] = [3^r,1]$ or $[1,3^r]$ in all possible combinations to produce $2^r$ different rows. Note that in each row the entry where all components of the form $3^r$ match is much larger than the rest. Hence $Q$ is diagonally dominant (when its rows are ordered suitably), and has full rank.


