The qtx(q+1)tordered orthogonal arrays (OOAs) of strength t over the alphabet Fq were constructed using linear feedback shift register sequences (LFSRs) defined by primitive polynomials in Fq[x]. In this paper, we extend this result to all polynomials in Fq[x] which satisfy some fairly simple restrictions, i.e., the 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 oft, in many cases, we still obtain the maximum number of columns in the constructed OOA when using non-primitive polynomials. For 2 ≤ q ≤ 9 and small t, we generate OOAs in this manner for all permissible polynomials of degree t in Fq[x] and compare the results to the ones produced in [2], [16], and [17] showing how close the arrays are to being “full” orthogonal arrays. Unusually for the finite fields, our arrays based on the non-primitive irreducible and even reducible polynomials are closer to the orthogonal arrays than those built from the primitive polynomials.