Good random matrices over finite fields
2012
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.
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