  We discuss a conjecture concerning the enumeration of nonsingular matrices
over a finite field that are block companion and whose order is the maximum
possible in the corresponding general linear group. A special case is proved
using some recent results on the probability that a pair of polynomials with
coefficients in a finite field is coprime. Connection with an older problem of
Niederreiter about the number of splitting subspaces of a given dimension are
outlined and an asymptotic version of the conjectural formula is established.
Some applications to the enumeration of nonsingular Toeplitz matrices of a
given size over a finite field are also discussed.
