Good random matrices over finite fields

Shengtian Yang,Thomas Honold

doi:10.3934/amc.2012.6.203

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.

Highlights

Let Fq be the finite field of order q and Gthe random matrix uniformly distributed over the set Fm q )/(f (Fm q) ×n of all m × n matrices over Fq
We prove two fundamental facts—a random matrix is good if and only if its transpose is good, and a random matrix uniformly distributed over an MRD code is good
We show that the minimum support size of a good random m × n matrix is qmax{m,n} and that a random m × n matrix with support size qmax{m,n} is good if and only if it is uniformly distributed over an (m, n, 1) MRD code

Summary

Introduction

Let Fq be the finite field of order q and Gthe random matrix uniformly distributed over the set Fm q ×n of all m × n matrices over Fq. In [34], it was shown that a random m × n matrix uniformly distributed over a Gabidulin code, a special kind of maximum-rank-distance (MRD) code [12, 15, 31], is a good random matrix with a distribution of support size as small as qmax{m,n}. What is the minimum achievable support size (of the distribution) of a good random matrix and what is the relation between good random matrices and MRD codes?.

Good Random Matrices and MRD Codes

Homogeneous Weights on Matrix Spaces

Applications of k-Good Random Matrices

Dense Sets of Matrices

Conclusion