## Base Reduction for Projective Spaces

Given a linear orthogonal array OA(*b*^{rm}, *s*,**F**_{br}, *k*) with *m* ≥ 1, a linear OA(*b*^{rm-r+1}, *s*,**F**_{b}, *k*) can be constructed. Correspondingly, a linear [*s*, *s*−(*m*−1)*r*−1, *d*]-code over **F**_{b} can be constructed based on a linear [*s*, *s*−*m*, *d*]-code over **F**_{br}.

### Construction

The new OA is constructed by interpreting the OAs as (multi)sets of points in the projective spaces PG(*m* – 1, *b*^{r}) and PG((*m*−1)*r*, *b*), respectively, and by noting that the first space can be embedded in the second one in a straightforward way.

Let A denote the OA(*b*^{rm}, *s*,**F**_{br}, *k*) and let * C* denote its generator matrix. Without loss of generality assume that the first row of

*has only entries 0 and 1 (otherwise multiply the columns of*

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

**C***represents a point in 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 OA(*

**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 OA(*

**C***b*

^{r(m−1)+1},

*s*,

**F**

_{b},

*k*) is obtained.

### See Also

If

*k*= 3 and if the points corresponding to A are all in AG(*m*– 1,*b*^{r}), this construction is the same as base reduction for affine capsCorresponding result for nets

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