The distribution of the rang of random matrices over a finite field

The present paper is concerned with a random matrix $A=\|a_{ij}\|$ ($i=\overline{1,t}$; $j=\overline{1,n}$), where $a_{ij}$ are independent random variables from a finite field $GF(q)$ with the following distribution:
$$ \mathbf P\{a_{ij}=a\in GF(q)\}= \begin{cases} 1-\frac{\ln e^xn}n,&\text{if}\quad a=0 \\ \frac{\ln e^xn}{(q-1)n},&\text{if}\quad a\ne0 \end{cases} $$
($x$ is a fixed number).
The distribution of the matrix rang for different values of $t$ and $n$ is found.