Base Reduction for Projective Spaces (for Nets)

Given a digital (mk, m, s)-net over Fbr with m ≥ 1, a digital (mʹ−k, mʹ, s)-net over Fb with mʹ = rmr + 1 can be constructed.

Construction

Let A denote the ordered orthogonal array OOA(bm, s,Fbr,∞, k) defined by the (mk, 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 C 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(m – 1, br) (the columns with a 1 in the first row are actually in the affine space AG(m – 1, br).

Now choose an arbitrary linear bijection φ : FbrFbr such that φ(1) = (0,…, 0, 1) and apply φ to each element of C such that an (rm)×(s,∞)-matrix Cʹ is obtained. It is easy to see that Cʹ is the generator matrix of a linear OOA(brm, s,Fb,∞, k). Since the first r−1 rows of Cʹ are zero, these rows can be discarded and a generator matrix of a linear OOA(b, s,Fb,∞, k) is obtained, which defines a digital (mʹ−k, mʹ, s)-net over Fb.

See Also

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

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