A General Construction of Ordered Orthogonal Arrays Using LFSRs.

The $q^{t} \times (q+1)t$ ordered orthogonal arrays (OOAs) of strength $t$ over the alphabet $ \mathbb {F}_{q}$ were constructed using linear feedback shift register sequences (LFSRs) defined by primitive polynomials in $ \mathbb {F}_{q}[x]$. In this paper, we extend this result to all polynomials in $\mathbb {F}_{q}[x]$ 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 of $t$ , in many cases, we still obtain the maximum number of columns in the constructed OOA when using non-primitive polynomials. For $2 \le q \le 9$ and small $t$ , we generate OOAs in this manner for all permissible polynomials of degree $t$ in $ \mathbb {F}_{q}[x]$ and compare the results to the ones produced in , , and 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. [ABSTRACT FROM AUTHOR]