Base Reduction for Projective Spaces

Given a linear orthogonal array OA(b^rm, s,F_b^r, 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_b^r.

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_b^r, k) and let C denote its generator matrix. Without loss of generality assume that the first row of C has only entries 0 and 1 (otherwise multiply the columns of C with suitable constants). Each column of C represents a point in 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 OA(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 OA(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 caps

• Corresponding 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: 2024-09-05. http://mint.sbg.ac.at/desc_CBRedPG.html