Weight and rank of matrices over finite fields
  Define the weight of a matrix to be the number of non-zero entries. One would like to count $m$ by $n$ matrices over a finite field by their weight and rank. This is equivalent to determining the probability distribution of the weight while conditioning on the rank. The complete answer to this question is far from finished. As a step in that direction this paper finds a closed form for the average weight of an $m$ by $n$ matrix of rank $k$ over the finite field with $q$ elements. The formula is a simple algebraic expression in $m$, $n$, $k$, and $q$. For rank one matrices a complete description of the weight distribution is given and a central limit theorem is proved. 
Theresa Migler, Kent E. Morrison, and Mitchell Ogle