Constructions of optimal orthogonal arrays with repeated rows

Abstract

We construct orthogonal arrays OA${}_{\lambda }(k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m/\lambda$ is as large as possible; these OAs are termed optimal. We provide constructions of optimal OAs for any $k\ge n+1$, albeit with large $\lambda$. We also study basic OAs; these are optimal OAs in which $\gcd (m,\lambda )=1$. We construct a basic OA with $n=2$ and $k=4t+1$, provided that a Hadamard matrix of order $8t+4$ exists. This completely solves the problem of constructing basic OAs wth $n=2$, modulo the Hadamard matrix conjecture.