Existence of small ordered orthogonal arrays

We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled.
11 pages; this revision contains a corrected version of Theorem 1, and small changes taking into account referee comments