## OOA Stacking with Additional Row

Let A denote a (linear) ordered orthogonal array OOA(M, s, Sb, T , k) with T ≥ ⌊k/2⌋. Then a (linear) OOA(M, s – 1, Sb, T ʹ, k) can be constructed for any T ʹ ≤ 3T . If M = bm and T is chosen as T = k (which is always possible), a (digital) (mk, m, s−1)-net in base b is obtained.

For T = 1 and k = 3 the result in the linear case is due to [1, Theorem 2] and [2, Theorem 1], the non-linear case to [1, Theorem 6] and [3, Theorem 6.2.1].

### Construction

The OOA Aʹ is constructed as follows based on A: Let σ denote a permutation of {1,…, sʹ} without fixed points. Then the ith factor of Aʹ is constructed as

(xi,1,…, xi,T ,  xs,1,…, xs,T ,  xσ(i), T,…, xσ(i), 1)

for i = 1,…, s−1 based on (x1,1,…, x1,T | ⋯ | xs,1,…, xs,T ) ∈ A.

Note that the first T levels of Aʹ are obtained by discarding a factor from A . The discarded factor of A is copied into all factors of Aʹ for the next T levels. Finally, the last T levels are a factor-wise permuted and level-wise reversed copy of the first T levels.