## NRT-Code Embedding in Larger Space

Every (linear) ordered orthogonal array OOA(*b*^{m}, *s*, *S*_{b}, *T *, *k*) yields a (linear) OOA(*b*^{m+T}, *s* + 1, *S*_{b}, *T *, *k*). Correspondingly, every (linear) ((*s*, *T *), *N*, *d*)-NRT-code yields a (linear) ((*s* + 1, *T *), *N*, *d*)-code over the same field.

Note that there is no corresponding propagation rule for nets, because *m*ʹ = *b* + *T * turns towards infinity if *T *→∞.

### Construction for OOAs

Based on a given OOA A the new orthogonal array Aʹ is obtained as

*,*

**x***) :*

**y***∈ A,*

**x***∈*

**y***S*

_{b}

^{T }}.

If * H* is a generator matrix of A, the generator matrix of Aʹ is given by

*ʹ = .*

**H**In other words, Aʹ is the direct product of A and the complete OOA OOA(*b*^{T }, 1, *S*_{b}, *T *, *T *).

### Construction for Linear NRT-Codes

The new linear NRT-code Cʹ is obtained by embedding C ⊆ **F**_{b}^{(s,T )} in **F**_{b}^{(s+1,T )}. In other words, Cʹ is constructed by appending a 0-block to every code word of C or by appending an all-zero block to the generator matrix of C.

### See Also

Special case for OAs / linear codes

### 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. “NRT-Code Embedding in Larger Space.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_OEmbeddingInLargerSpace.html