## (*u*, *u*+*v*)-Construction for OOAs

Given two (linear) ordered orthogonal arrays OOA(*M*_{i}, *s*_{i}, *S*_{b}, *T *, *k*_{i}) with *s*_{1} ≤ *s*_{2}, a (linear) OOA(*M*_{1}*M*_{2}, *s*_{1} + *s*_{2}, *S*_{b}, *T *, *k*) with *k* = min{2*k*_{1} +1, *k*_{2}} can be constructed [1, Theorem 5.1].

Correspondingly, given two (linear) ((*s*_{i}, *T *), *N*_{i}, *d*_{i})-NRT-codes over the same field with *s*_{1} ≤ *s*_{2}, a (linear) ((*s*_{1} + *s*_{2}, *T *), *N*_{1}*N*_{2}, *d*)-code with *d* = min{2*d*_{1}, *d*_{2}} can be constructed [1, Theorem 5.2].

### The Construction for OOAs

Let A_{1} and A_{2} denote the OOAs with parameters OOA(*M*_{1}, *s*_{1}, *S*_{b}, *T *, *k*_{1}) and OOA(*M*_{2}, *s*_{2}, *S*_{b}, *T *, *k*_{2}), respectively, and let *s*_{1} ≤ *s*_{2}. Then the resulting OOA A with parameters OOA(*M*_{1}*M*_{2}, *s*_{1} + *s*_{2}, *S*_{b}, *T *, *k*) is given by

*–*

**u***π*(

*),*

**v***) :*

**v***∈ A*

**u**_{1},

*∈ A*

**v**_{2}}

where *π* : *S*_{b}^{(s2, T)}→*S*_{b}^{(s1, T)} is the projection selecting the first *s*_{1} factors.

If A_{1} and A_{2} are linear with *M*_{i} = *b*^{mi} and *m*_{i}×(*s*_{i}, *T *) generator matrices **H**_{i}, the (*m*_{1} + *m*_{2})×(*s*_{1} + *s*_{2}, *T *)generator matrix of A is given by

with *π*(**H**_{2}) denoting the first *s*_{1} factors of **H**_{2}.

### The Construction for NRT-Codes

Let C_{1} be an ((*s*_{1}, *T *), *N*_{1}, *d*_{1})-code and let C_{2} be an ((*s*_{2}, *T *), *N*_{2}, *d*_{2})-code, both over **F**_{b} with *s*_{1} ≤ *s*_{2}. Then the set of vectors

*,(*

**u***, 0*

**u**_{n1×(s2-s1, T)}) +

*) :*

**v***∈ C*

**u**_{1},

*∈ C*

**v**_{2}}

is an ((*s*_{1} + *s*_{2}, *T *), *N*_{1}*N*_{2}, *d*)-code over **F**_{b} with *d* = min{2*d*_{1}, *d*_{2}}.

If C_{1} and C_{2} are linear with *N*_{i} = *b*^{ni}, *m*_{i} = *Ts*_{i} – *n*_{i}, *n*_{i}×(*s*, *T *) generator matrices **G**_{i}, and *m*_{i}×(*s*_{i}, *T *) parity check matrices **H**_{i}, the generator matrix of the new linear [(*s*_{1} + *s*_{2}, *T *), *n*_{1} + *n*_{2}, *d*]-code C is given by

its parity check matrix is shown above.

### A Special Case

If an OOA(*b*^{m2}, *s*_{2}, *S*_{b}, *T *, 3) is combined with the OOA(*b*, *s*_{2}, *S*_{b}, *T *, 1) obtained from embedding the parity-check code (which exists for all *s* ≥ 1), applying the (*u*, *u* + *v*)-construction gives an OOA(*b*^{m2+1}, 2*s*, *S*_{b}, *T *, 3), and (after *u* applications) an OOA(*b*^{m2+u}, *us*, *S*_{b}, *T *, 3) all *u* ≥ 0. This case is treated separately in MinT in order to list a single rule application instead of the full stack.

### See Also

Special case for orthogonal arrays / codes

For

*T*→∞ the corresponding result for nets is obtainedA weaker construction yielding NRT-codes and OOAs of the same size, but with smaller minimum distance / strength is the direct product, which can be seen as a “(

*u*,*v*)-construction”.

### 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 |

### 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. “(*u*, *u*+*v*)-Construction for OOAs.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_OUUPlusV.html