On orthogonal arrays of strength 3 and 5 achieving Rao's bound

It is shown that ifA is an orthogonal array (N, n, q, 3) achieving Rao's bound, thenA is either

1. an orthogonal array (2n, n, 2, 3) withn ≡ 0 (mod 4), or
2. an orthogonal array (q ³,q + 2,q, 3) withq even.

This result should be compared with a theorem of P.J. Cameron on extendable symmetric designs. It is also shown that ifA is an orthogonal array (N, n, q, 5) achieving Rao's bound, thenA is either the orthogonal array (32, 6, 2, 5) or the orthogonal array (3⁶, 12, 3, 5).