Embedding of orthogonal arrays of strength two and deficiency greater than two

Abstract

Let x≥0 and n≥2 be integers. Suppose there exists an orthogonal array A(n, q, μ^∗) of strength 2 in n symbols with q rows and n²μ^∗ columns where q^∗=q−d, q^∗=n²x+n+1, μ^∗=(n−1)x+1 and d is a positive integer. Then d is called the deficiency of the orthogonal array. The question of embedding such an array into a complete array A(n, q^∗, μ^∗) is considered for the case d≥3. It is shown that for d=3 such an embedding is always possible if n≥2(d−1)²(2d²−2d+1). Partial results are indicated if d≥4 for the embedding of a related design in a corresponding balanced incomplete block design.