## Discarding Factors for OOAs / Shortening the Dual NRT-Code

Every (linear) ordered orthogonal array OOA(*M*, *s*, *S*_{b}, *T *, *k*) yields a (linear) OOA(*M*, *s* – 1, *S*_{b}, *T *, *k*). Correspondingly, every linear [(*s*, *T *), *n*, *d*]-NRT-code with *n* ≥ *T * yields a linear [*s*−1, *n*−*T *, *d*]-code over the same field.

### The Construction for OOAs

The new OOA Aʹ is obtained by discarding an arbitrary factor from the original array A, i.e., as

*x*

_{i,1},…,

*x*

_{i,T })

_{i=2,…, s}:

*∈ A}.*

**x**If A is linear, a generator matrix of Aʹ is obtained by dropping an arbitrary block from the generator matrix of A.

### The Construction for Linear NRT-Codes

In the context of linear NRT-codes, this propagation rule corresponds to shortening a code: If C is the original code, the new code Cʹ is given by

*x*

_{i,1},…,

*x*

_{i,T })

_{i=2,…, s}:

*∈ C with*

**x***x*

_{1,1}= …

*x*

_{1,T }= 0},

i.e., it is obtained by selecting only code words with zeros in a fixed block and removing this block. Since all remaining code words have their weights unchanged, the minimum weight (and therefore the minimum distance) of Cʹ is unaffected. In the fortuitous case that one or more *j* ∈ {1,…, *T *} exist with *x*_{1,j} = 0 for all * x* ∈ C, Cʹ is actually an [

*s*−1,

*n*ʹ,

*d*]-code with

*n*ʹ >

*n*−

*T*and the final [

*s*−1,

*n*−

*T*,

*d*]-code can be obtained by taking a subcode.

### See Also

Special case for OAs / linear codes

For

*T*→∞ one obtains the corresponds result for nets.Corresponding result for sequences

Discarding a different set of columns leads to the propagation rule depth reduction

### 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. “Discarding Factors for OOAs / Shortening the Dual NRT-Code.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_OSRed.html