The random matrix uniformly distributed over the set of all mby-n matrices over a finite field plays an important role in many branches of information theory. In this paper a generalization of this random matrix, called k-good random matrices, is studied. It is shown that a k-good random m-by-n matrix with a distribution of minimum support size is uniformly distributed over a maximum-rank-dista...

Matrices with very few nonzero entries cannot have large rank. On the other hand matrices without any zero entries can have rank as low as 1. These simple observations lead us to our main question. For matrices over finite fields, what is the relationship between the rank of a matrix and the number of nonzero entries in the matrix? This question motivated a summer research project collaboration...