Trace Code for OOAs

Let A be a (linear) ordered orthogonal array OOA(M, s, Sbu, T , k) with u ≥ 1. Then a (linear) OOA(M, us, Sb, T , k) Aʹ can be constructed.

Without loss of generality let Sbu := {0,…, bu – 1} and Sb := {0,…, b – 1}. Define φi : SbuSb such that x = $ \sum_{{i=1}}^{{u}}$φi(x)bi−1 for all xSbu. If b is a prime power, we can also choose Sbu := Fbu, Sb := Fb, and φi(x) := tr(αix) where {α1,…, αu} is a basis of Fbu over Fb and tr : FbuFb is the trace. Therefore this construction usually goes by the name trace code.

Now Aʹ is defined as the multi-set

Aʹ = {(φ1(x),…, φu(x))  :  xA} ⊆ Fb(us,T )

with φi applied coordinate-wise to the vectors in Fbu(s,T ).

Since the trace is linear, this construction also yields a linear OOA(bum, us,Fb, T , k) if C is a linear OOA(bum, s,Fbu, T , k): Let H be the m×(s, T ) generator matrix of C over Fbu. Then

$\displaystyle \left(\vphantom{\begin{array}{ccc} \varphi_{1}(\alpha_{1}\vec{H})… …\alpha_{u}\vec{H}) & \cdots & \varphi_{u}(\alpha_{u}\vec{H})\end{array}}\right.$$\displaystyle \begin{array}{ccc} \varphi_{1}(\alpha_{1}\vec{H}) & \cdots & \var… …phi_{1}(\alpha_{u}\vec{H}) & \cdots & \varphi_{u}(\alpha_{u}\vec{H})\end{array}$$\displaystyle \left.\vphantom{\begin{array}{ccc} \varphi_{1}(\alpha_{1}\vec{H})… …\alpha_{u}\vec{H}) & \cdots & \varphi_{u}(\alpha_{u}\vec{H})\end{array}}\right)$

with φi applied again element-wise is an um×(us, T ) generator matrix of Cʹ over Fb.

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

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: 2008-04-04. http://mint.sbg.ac.at/desc_OTrace.html

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