Rank of a tall binary matrix with IID Bernoulli entries with probability function of size. I have a binary matrix composed of $0's$ and $1's$. $\textbf{X}$ of size $(n,k)$ where $n>k$.
The entries of the matrix are Bernoulli random variables. Specifically, the $(i,j)^{th}$ entry $X_{ij} \sim Bern(q), \forall i,j.$ Also, $X_{ij} \in \{0,1\}$. Note that  $X_{ij}$ are independent  random variables $\forall i,j.$
I am operating on the field $\mathbb{R}$.
Question: $\boxed{\text{What is the probability that Rank(X)=$k$  if $q=1-2^{-1/k}$.}}$
Moreover, in my case, $\boxed{n= C \times k \log k \text{ where C>0 is a constant such that $n>k$.}} $
Can someone provide a  characterization of $Rank(\textbf{X})$ in terms of $q$ and $n$.
Are there results in the literature which talks about "tall Bernoulli matrices" with probability as a function of the matrix dimension? Tight bounds are ok too.

The context: Estimating parameters using 0-1 matrix system of equations
To find the least squares estimate, I need to find the psuedo-inverse
of $\textbf{X}$. The least squares estimate is only a good one, if the
matrix $\textbf{X}$ is  non-singular and well-conditioned.

 A: Assume the entries are i.i.d. Bernoulli with $P[X_{ij}=1]=1-(1/2)^{1/k}$, where $k$ is the number of columns. Assume the number of rows is $n=c k \log(k)$ for some $c>0$.
The probability that a single column is all-zero is
$$ P[\mbox{Col 1 is all-zero}]=(1/2)^{n/k} = (1/2)^{c \log(k)} = \frac{1}{k^{c\log(2)}}$$
The probability that no columns are all-zero is:
$$ P[\mbox{No columns are all-zero}]=\left(1 - \frac{1}{k^{c\log(2)}}\right)^k$$
Thus
$$ \lim_{k\rightarrow\infty} P[\mbox{No columns are all-zero}] = \left\{\begin{array}{cc}
0 & \mbox{ if $0<c<\frac{1}{\log(2)}$} \\
1/e & \mbox{ if $c=\frac{1}{\log(2)}$}\\
1 & \mbox{ if $c>\frac{1}{\log(2)}$} 
\end{array}\right.$$
where $\frac{1}{\log(2)} \approx 1.442695$. If a matrix has at least one all-zero column then it cannot be full rank.  Thus, assuming $0<c<\frac{1}{\log(2)}$ we have
$$ \lim_{k\rightarrow\infty} P[\mbox{Full rank}] = 0$$
The case $c>\frac{1}{\log(2)}$ can perhaps make use of the approximations here:
https://mathoverflow.net/questions/330669/rank-of-a-random-sparse-matrix-with-nonnegative-reals
