  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.
