NRT-Code Embedding in Larger Space

Every (linear) ordered orthogonal array OOA(bm, s, Sb, T , k) yields a (linear) OOA(bm+T, s + 1, Sb, 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 mʹ = b + T turns towards infinity if T →∞.

Construction for OOAs

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

Aʹ = {(x,y)  :  x ∈ A,y ∈ SbT }.

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

Hʹ = $\displaystyle \left(\vphantom{\begin{array}{cc} \vec{H} & \vec{0}_{b^{m}\times T}\\ \vec{0}_{T\times(s,T)} & \vec{1}_{T}\end{array}}\right.$$\displaystyle \begin{array}{cc} \vec{H} & \vec{0}_{b^{m}\times T}\\ \vec{0}_{T\times(s,T)} & \vec{1}_{T}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cc} \vec{H} & \vec{0}_{b^{m}\times T}\\ \vec{0}_{T\times(s,T)} & \vec{1}_{T}\end{array}}\right)$.

In other words, Aʹ is the direct product of A and the complete OOA OOA(bT , 1, Sb, T , T ).

Construction for Linear NRT-Codes

The new linear NRT-code Cʹ is obtained by embedding C ⊆ Fb(s,T ) in Fb(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

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

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