Geodesic
The distribution of the rang of random matrices over a finite field
Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 4, pp. 631-641 
G. V. Balakin. The distribution of the rang of random matrices over a finite field. Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 4, pp. 631-641. http://geodesic.mathdoc.fr/item/TVP_1968_13_4_a3/
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     author = {G. V. Balakin},
     title = {The distribution of the rang of random matrices over a~finite field},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {631--641},
     year = {1968},
     volume = {13},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1968_13_4_a3/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.