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In \cite{Castoldi}, $q^t \by (q+1)t$ ordered orthogonal arrays (OOAs) ofstrength $t$ over the alphabet $\FF_q$ were constructed using linear feedbackshift 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 areautomatically satisfied by primitive polynomials. While these restrictionssometimes reduce the number of columns produced from $(q+1)t$ to a smallermultiple of $t$, in many cases we still obtain the maximum number of columns inthe constructed OOA when using non-primitive polynomials. For small values of$q$ and $t$, we generate OOAs in this manner for all permissible polynomials ofdegree $t$ in $\FF_q[x]$ and compare the results to the ones produced in\cite{Castoldi}, \cite{Rosenbloom} and \cite{Skriganov} showing how close thearrays are to being "full" orthogonal arrays. Unusually for finite fields, ourarrays based on non-primitive irreducible and even reducible polynomials arecloser to orthogonal arrays than those built from primitive polynomials.

A general construction of Ordered Orthogonal Arrays using LFSRs