## OOA Folding and Stacking with Additional Row

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

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

### Construction

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

 ( xi+0sʹ,1,…, xi+0sʹ,T ,  …,  xi+(u−1)sʹ,1,…, xi+(u−1)sʹ,T , xs,1,…, xs,T , xσ(i)+(u−1)sʹ, 1,…, xσ(i)+(u−1)sʹ, T,  …,  xσ(i)+0sʹ, 1,…, xσ(i)+0sʹ, T )

for i = 1,…, sʹ. Note that the first Tu levels of Aʹ are obtained by u-times folding the first sʹu factors of A. The last factor of A is copied into all factors of Aʹ in levels Tu + 1,…,(u + 1)T . Finally, levels (u + 1)T + 1,…,(2u + 1)T are a factor-wise permuted and level-wise reversed copy of the first Tu levels.