Removing Affine Subspaces

Let m2(u, b) and Cu(b) denote the size of the largest caps in the projective space PG(u, b) and in the affine space AG(u, b), respectively. From now on we assume that b > 2 is a prime power and u ≥ 3.

Let H be a hyperplane in PG(u, b). Then H is isomorphic to PG(u − 1, b) and PG(u, b) ∖ H is isomorphic to AG(u, b). Therefore, every s-cap in PG(u, b) yields an s1-cap in PG(u − 1, b) and an s2-cap in AG(u, b) with s1 + s2 = s. Hence, we have

m2(u, b) ≤ m2(u – 1, b) + Cu(b)

and, by iterating this process,

m2(u, b) ≤ m2(uʹ, b) + $\displaystyle \sum_{{i=uʹ+1}}^{{u}}$Ci(b)

for all uʹ with 2 ≤ uʹ < u.


Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Removing Affine Subspaces.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03.

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