Ordered Orthogonal Array Construction Using LFSR Sequences

TL;DR

This paper introduces a novel method for constructing ordered orthogonal arrays of strength t using LFSR sequences, which are shown to be stronger and more closely resemble full orthogonal arrays than previous constructions.

Contribution

The paper presents a new construction of ordered orthogonal arrays using LFSR sequences, differing from prior methods and demonstrating improved properties.

Findings

Constructed OOAs match parameters of previous methods but differ in structure.

Experimental results show the new OOAs are closer to full orthogonal arrays.

The construction relates to and improves upon existing techniques using linear independence and hypergraph homomorphisms.

Abstract

We present a new construction of ordered orthogonal arrays (OOA) of strength $t$ with $(q + 1)t$ columns over a finite field $\mathbb{F}_{q}$ using linear feedback shift register sequences (LFSRs). OOAs are naturally related to $(t, m, s)$-nets, linear codes, and MDS codes. Our construction selects suitable columns from the array formed by all subintervals of length $\frac{q^{t}-1}{q-1}$ of an LFSR sequence generated by a primitive polynomial of degree $t$ over $\mathbb{F}_{q}$. We prove properties about the relative positions of runs in an LFSR which guarantee that the constructed OOA has strength $t$. The set of parameters of our OOAs are the same as the ones given by Rosenbloom and Tsfasman (1997) and Skriganov (2002), but the constructed arrays are different. We experimentally verify that our OOAs are stronger than the Rosenbloom-Tsfasman-Skriganov OOAs in the sense that ours are "closer" to being a "full" orthogonal array. We also discuss how our OOA construction relates to previous techniques to build OOAs from a set of linearly independent vectors over $\mathbb{F}_{q}$, as well as to hypergraph homomorphisms.