A general construction of Ordered Orthogonal Arrays using LFSRs

TL;DR

This paper generalizes the construction of ordered orthogonal arrays using LFSRs from primitive polynomials to all polynomials satisfying certain conditions, often producing arrays closer to ideal orthogonal arrays.

Contribution

It extends previous methods to include all polynomials with simple restrictions, often maximizing columns and improving array quality.

Findings

Arrays based on non-primitive polynomials are closer to true orthogonal arrays.

The method produces arrays with maximum columns in many cases.

Comparison shows improved array quality over prior constructions.

Abstract

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.