## Base Reduction for Projective Spaces (for Nets)

Given a digital (*m*−*k*, *m*, *s*)-net over **F**_{br} with *m* ≥ 1, a digital (*m*ʹ−*k*, *m*ʹ, *s*)-net over **F**_{b} with *m*ʹ = *rm*−*r* + 1 can be constructed.

### Construction

Let A denote the ordered orthogonal array OOA(*b*^{m}, *s*,**F**_{br},∞, *k*) defined by the (*m*−*k*, *m*, *s*)-net. Let * C* denote the

*m*×(

*s*,∞) generator matrix of A. Without loss of generality assume that the entries in the first row of

*are either 0 or 1 (otherwise multiply the columns of*

**C***with suitable constants). Each column of*

**C***represents a point in the projective space PG(*

**C***m*– 1,

*b*

^{r}) (the columns with a 1 in the first row are actually in the affine space AG(

*m*– 1,

*b*

^{r}).

Now choose an arbitrary linear bijection *φ* : **F**_{br}→**F**_{b}^{r} such that *φ*(1) = (0,…, 0, 1) and apply *φ* to each element of * C* such that an (

*rm*)×(

*s*,∞)-matrix

*ʹ is obtained. It is easy to see that*

**C***ʹ is the generator matrix of a linear OOA(*

**C***b*

^{rm},

*s*,

**F**

_{b},∞,

*k*). Since the first

*r*−1 rows of

*ʹ are zero, these rows can be discarded and a generator matrix of a linear OOA(*

**C***b*

^{mʹ},

*s*,

**F**

_{b},∞,

*k*) is obtained, which defines a digital (

*m*ʹ−

*k*,

*m*ʹ,

*s*)-net over

**F**

_{b}.

### See Also

Corresponding result for orthogonal arrays

Corresponding result for affine caps

### 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. “Base Reduction for Projective Spaces (for Nets).”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_NBRedPG.html