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

Let x ≥ 0 and n ≥ 2 be integers. Suppose there exists an orthogonal array $\text{A(n, q, μ}{}^1\text{)}$ of strength 2 in n symbols with q rows and $\text{n}{}^2\text{μ}{}^1$ columns where $\text{q = q}{}^1\text{? d, q}{}^1\text{= n}{}^2\text{x + n + 1}$, $\text{μ}{}^1\text{= (n ?}\text{1}\text{)x +}\text{1}$ 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 $\text{A(n, q}{}^1\text{, μ}{}^1\text{)}$ 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.