## Trace Code for OOAs

Let A be a (linear) ordered orthogonal array OOA(*M*, *s*, *S*_{bu}, *T *, *k*) with *u* ≥ 1. Then a (linear) OOA(*M*, *us*, *S*_{b}, *T *, *k*) Aʹ can be constructed.

Without loss of generality let *S*_{bu} := {0,…, *b*^{u} – 1} and *S*_{b} := {0,…, *b* – 1}. Define *φ*_{i} : *S*_{bu}→*S*_{b} such that *x* = *φ*_{i}(*x*)*b*^{i−1} for all *x* ∈ *S*_{bu}. If *b* is a prime power, we can also choose *S*_{bu} := **F**_{bu}, *S*_{b} := **F**_{b}, and *φ*_{i}(*x*) := tr(*α*_{i}*x*) where {*α*_{1},…, *α*_{u}} is a basis of **F**_{bu} over **F**_{b} and tr : **F**_{bu}→**F**_{b} is the trace. Therefore this construction usually goes by the name trace code.

Now Aʹ is defined as the multi-set

*φ*

_{1}(

*),…,*

**x***φ*

_{u}(

*)) :*

**x***∈ A} ⊆*

**x****F**

_{b}

^{(us,T )}

with *φ*_{i} applied coordinate-wise to the vectors in **F**_{bu}^{(s,T )}.

Since the trace is linear, this construction also yields a linear OOA(*b*^{um}, *us*,**F**_{b}, *T *, *k*) if C is a linear OOA(*b*^{um}, *s*,**F**_{bu}, *T *, *k*): Let * H* be the

*m*×(

*s*,

*T*) generator matrix of C over

**F**

_{bu}. Then

with *φ*_{i} applied again element-wise is an *um*×(*us*, *T *) generator matrix of Cʹ over **F**_{b}.

For linear OOAs this result is due to [1, Theorem 3.1]. The special case for *T * = *k* is given in [2, Propagation Rule 7].

### See Also

Special case for OAs / linear codes

For

*T*→∞ one obtains the corresponds result for nets

### 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. Constructions of ( t, m, s)-nets.In Harald Niederreiter and Jerome Spanier, editors, Monte Carlo and Quasi-Monte Carlo Methods 1998, pages 70–85. Springer-Verlag, 2000. |

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