Good Random Matrices over Finite Fields

TL;DR

This paper introduces and studies k-good random matrices over finite fields, linking their distributions to MRD codes and exploring their combinatorial properties and applications in information theory.

Contribution

It generalizes the concept of random matrices over finite fields to k-good matrices, establishing their relation to MRD codes and analyzing their combinatorial structures.

Findings

k-good matrices are uniformly distributed over MRD codes of specific minimum rank distance

Examples derived from homogeneous weights on matrix modules

Minimum size of k-dense sets characterized in certain cases

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 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-distance (MRD) code of minimum rank distance min{m,n}-k+1, and vice versa. Further examples of k-good random matrices are derived from homogeneous weights on matrix modules. Several applications of k-good random matrices are given, establishing links with some well-known combinatorial problems. Finally, the related combinatorial concept of a k-dense set of m-by-n matrices is studied, identifying such sets as blocking sets with respect to (m-k)-dimensional flats in a certain m-by-n matrix geometry and determining their minimum size in special cases.