Base Reduction for Projective Spaces

Given a linear orthogonal array OA(brm, s,Fbr, k) with m ≥ 1, a linear OA(brm-r+1, s,Fb, k) can be constructed. Correspondingly, a linear [s, s−(m−1)r−1, d]-code over Fb can be constructed based on a linear [s, sm, d]-code over Fbr.

Construction

The new OA is constructed by interpreting the OAs as (multi)sets of points in the projective spaces PG(m – 1, br) 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(brm, s,Fbr, 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, 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 (rms-matrix Cʹ is obtained. It is easy to see that Cʹ is the generator matrix of a linear OA(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 OA(br(m−1)+1, s,Fb, k) is obtained.

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.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_CBRedPG.html

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