  In \cite{Castoldi}, $q^t \by (q+1)t$ ordered orthogonal arrays (OOAs) of
strength $t$ over the alphabet $\FF_q$ were constructed using linear feedback
shift register sequences (LFSRs) defined by {\em primitive} polynomials in
$\FF_q[x]$. In this paper we extend this result to all polynomials in
$\FF_q[x]$ which satisfy some fairly simple restrictions, restrictions that are
automatically satisfied by primitive polynomials. While these restrictions
sometimes reduce the number of columns produced from $(q+1)t$ to a smaller
multiple of $t$, in many cases we still obtain the maximum number of columns in
the constructed OOA when using non-primitive polynomials. For small values of
$q$ and $t$, we generate OOAs in this manner for all permissible polynomials of
degree $t$ in $\FF_q[x]$ and compare the results to the ones produced in
\cite{Castoldi}, \cite{Rosenbloom} and \cite{Skriganov} showing how close the
arrays are to being "full" orthogonal arrays. Unusually for finite fields, our
arrays based on non-primitive irreducible and even reducible polynomials are
closer to orthogonal arrays than those built from primitive polynomials.
