A Note on Good Random Matrices over Finite Fields

Abstract

The random matrix uniformly distributed over the set of all m-by-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 good random matrices, is studied. It is shown that a good random m-by-n matrix with a distribution of minimum support size is a random matrix uniformly distributed over a maximum-rank-distance (MRD) code with rank distance min{m,n}, and vice versa. Furthermore, a special family of sets of matrices, called complete subspaces, is investigated. It is shown that the random matrix uniformly distributed over a complete subspace is good. Some fundamental properties of complete subspaces are presented.