Base Reduction for Projective Spaces (for Nets)

Given a digital (m−k, m, s)-net over F_b^r with m ≥ 1, a digital (mʹ−k, mʹ, s)-net over F_b with mʹ = rm−r + 1 can be constructed.

Construction

Let A denote the ordered orthogonal array OOA(b^m, s,F_b^r,∞, 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 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, 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_b^r→F_b^r 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(b^rm, s,F_b,∞, 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^mʹ, s,F_b,∞, k) is obtained, which defines a digital (mʹ−k, mʹ, 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: 2024-09-05. http://mint.sbg.ac.at/desc_NBRedPG.html