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

G. V. Balakin

Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 4, pp. 631-641

Résumé

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.