## Truncation for OOAs

Every (linear) ordered orthogonal array OOA(*b*^{m}, *s*, *S*_{b}, *T *, *k*) yields a (linear) OOA(*b*^{m−u}, *s* – 1, *S*_{b}, *T *, *k*−*u*) 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 *), *n*−*T * + *u*, *d*−*u*]-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*, *d*−*T *)-code over the same field.

### Construction for OOAs

The new OOA Aʹ is obtained from A as

*x*

_{i,1},…,

*x*

_{i,T })

_{i=2,…, s}:

*∈ A and*

**x***x*

_{1,1}= … =

*x*

_{1,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 (*m* – *u*)×(*s*−1, *T *)-matrix * H*ʹ is a generator matrix of Aʹ. For the special case

*u*=

*T*the matrix

*is partitioned as*

**H***= .*

**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

*x*

_{i,1},…,

*x*

_{i,T })

_{i=2,…, s}:

*∈ C}*

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

### See Also

Special case for orthogonal arrays and linear codes

### References

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

### Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.

Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Truncation for OOAs.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_OMRedS.html