## Truncation for OOAs

Every (linear) ordered orthogonal array OOA(bm, s, Sb, T , k) yields a (linear) OOA(bmu, s – 1, Sb, T , ku) for all u with 1 ≤ u ≤ min{T , k} [1, Corollary 5.9].

Correspondingly, every linear [(s, T ), n, d]-NRT-code yields a linear [(s−1, T ), nT + u, du]-code over the same field for all u with 1 ≤ u ≤ min{T , d – 1}. If u = T it can even be shown that every (not necessarily linear) ((s, T ), N, d)-code yields an ((s−1, T ), N, dT )-code over the same field.

### Construction for OOAs

The new OOA Aʹ is obtained from A as

Aʹ = {(xi,1,…, xi,T )i=2,…, s  :  xA and x1,1 = … = x1,u = 0},

i.e., by dropping one block and all runs that do not have certain fixed elements in the first u positions of this block. If H is a generator matrix of A in the form

H =   (which can always be achieved using elementary row operations), the (mu)×(s−1, T )-matrix Hʹ is a generator matrix of Aʹ. For the special case u = T the matrix H is partitioned as

H =   .

### Construction for NRT-Codes

For u < T the construction cannot be applied directly in NRT-space.

For u = T the new code Cʹ is constructed by dropping one block from all code words of C, i.e., as

Cʹ = {(xi,1,…, xi,T )i=2,…, s  :  xC}

and a generator matrix of Cʹ can be obtained by dropping an arbitrary block from a generator matrix of C.

  Rudolf Schürer.Ordered Orthogonal Arrays and Where to Find Them.PhD thesis, University of Salzburg, Austria, August 2006. 