## Concatenation of Two NRT-Codes

Given a linear [(*s*_{1}, *T*_{1}), *n*_{1}, *d*_{1}]-NRT-code C_{1} over **F**_{bn2} and a linear [(*s*_{2}, *T*_{2}), *n*_{2}, *d*_{2}]-NRT-code C_{2} over **F**_{b}, a linear [(*s*_{1}*s*_{2}, *T*_{1}*T*_{2}), *n*_{1}*n*_{2}, *d*_{1}*d*_{2}]-NRT-code over **F**_{b} can be constructed. Using duality, a linear ordered orthogonal array OOA(*b*^{s1s2T1T2-n1n2}, *s*_{1}*s*_{2},**F**_{b}, *T*_{1}*T*_{2}, *d*_{1}*d*_{2} − 1) can be constructed based on a linear OOA(*b*^{s1T1-m1}, *s*_{1},**F**_{bs2T2-n2}, *d*_{1} − 1) and OOA(*b*^{s2T2-m2}, *s*_{2},**F**_{b}, *T*_{2}, *d*_{2} − 1).

### Construction

Let *φ* : **F**_{bn2}↔C_{2} ⊆ **F**_{b}^{(s2, T2)} denote an arbitrary **F**_{b}-linear bijection. Then the new code C is defined as

*φ*(

*x*

_{i}))

_{(i, j) ∈ {1,…, s1}×{1,…, T1}}:

*∈ C*

**x**_{1}}

with the resulting indices for depth ordered first by *j* and secondly by the depth-index from C_{2}. More formally and assuming that the columns of C_{h} are indexed by {0,…, *s*_{h} – 1}×{0,…, *T*_{h} – 1} for *h* = 1, 2, C is defined as

*φ*(

*x*

_{⌊i/s2⌋,⌊j/T2⌋}))

_{i mod s2, j mod T2})

_{(i, j) ∈ {0,…, s1s2−1}×{0,…, T1T2−1}}:

*∈ C*

**x**_{1}}.

### See Also

Special case for 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. “Concatenation of Two NRT-Codes.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_OConcatD.html