## Concatenation of Two OOAs

Given two (linear) ordered orthogonal arrays A_{1}, A_{2} with parameters OOA(*M*_{1}, *s*_{1}, *S*_{M2}, *T*_{1}, *k*) and OOA(*M*_{2}, *s*_{2}, *S*_{b}, *T*_{2}, *k*), a new (linear) OOA(*M*_{1}, *s*_{1}*s*_{2}, *S*_{b}, *T*_{1}*T*_{2}, *k*) can be constructed [1, Theorem 4.1]. Using duality, a linear [(*s*_{1}*s*_{2}, *T*_{1}*T*_{2}), *s*_{1}*s*_{2}*T*_{1}*T*_{2} – *m*_{1}*m*_{2}, *k* + 1]-NRT-code over **F**_{b} can be constructed based on a linear [(*s*_{1}, *T*_{1}), *s*_{1}*T*_{1} – *m*_{1}, *k* + 1]-NRT-code over **F**_{bm2} and a linear [(*s*_{2}, *T*_{2}), *s*_{2}*T*_{2} – *m*_{2}, *k* + 1]-NRT-code over **F**_{b}.

### Construction

Let *φ* : *S*_{M2}↔A_{2} ⊆ *S*_{b}^{(s2, T2)} denote an arbitrary bijection. If A_{1} and A_{2} are linear, let *φ* : **F**_{bm2}↔A_{2} be **F**_{b}-linear. Then the resulting OOA A is defined as

*φ*(

*x*

_{i}))

_{(i, j) ∈ {1,…, s1}×{1,…, T1}}:

*∈ A*

**x**_{1}}

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

*φ*(

*x*

_{⌊i/s2⌋,⌊j/T2⌋}))

_{i mod s2, j mod T2})

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

*∈ A*

**x**_{1}}.

### Application to Nets

If either A_{1} or A_{2} has depth 1 (i.e., it is an orthogonal array) and the other one has depth *k*, A has also depth equal to strength. Since OOAs with depth equal to strength are equivalent to (*m*−*k*, *m*, *s*)-nets, this procedure can be used for obtaining a new net based on an orthogonal array and a net. In particular, we have the following two corollaries:

Given a (digital) (

*m*_{net}–*k*,*m*_{net},*s*_{net})-net in base*M*_{OA}and a (linear) OA(*M*_{OA},*s*_{OA},*S*_{b},*k*), a (digital) (*m*_{net}*m*_{OA}–*k*,*m*_{net}*m*_{OA},*s*_{net}*s*_{OA})-net in base*b*can be constructed (established for the digital case in [2, Threorem 11]).Given a (linear) OA(

*M*_{OA},*s*_{OA},*S*_{bmnet},*k*) and a (digital) (*m*_{net}–*k*,*m*_{net},*s*_{net})-net in base*b*, a (digital) (*m*_{net}*m*_{OA}–*k*,*m*_{net}*m*_{OA},*s*_{net}*s*_{OA})-net in base*b*can be constructed (established for the digital case in [2, Threorem 12]).

### See Also

Special case for orthogonal arrays

### References

[1] | Jürgen Bierbrauer, Yves Edel, and Wolfgang Ch. Schmid. Coding-theoretic constructions for ( t, m, s)-nets and ordered orthogonal arrays.Journal of Combinatorial Designs, 10(6):403–418, 2002.doi:10.1002/jcd.10015 |

[2] | Harald Niederreiter and Chaoping Xing. Nets, ( t, s)-sequences, and algebraic geometry.In Peter Hellekalek and Gerhard Larcher, editors, Random and Quasi-Random Point Sets, volume 138 of Lecture Notes in Statistics, pages 267–302. Springer-Verlag, 1998. |

### 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 OOAs.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_OConcatK.html