Journal of Geometry

, Volume 85, Issue 1–2, pp 35–41 | Cite as

Caps with free pairs of points

Original Paper

Abstract.

We say that two points x, y of a cap C form a free pair of points if any plane containing x and y intersects C in at most three points. For given N and q, we denote by m2+ (N, q) the maximum number of points in a cap of PG(N, q) that contains at least one free pair of points. It is straightforward to prove that m2+ (N, q) ≤ (qN-1  +  2q  −  3)/(q  − 1), and it is known that this bound is sharp for q = 2 and all N. We use geometric constructions to prove that this bound is sharp for all q when N ≤ 4. We briefly survey the motivation for constructions of caps with free pairs of points which comes from the area of statistical experimental design.

Mathematics Subject Classification (2000).

51E22 62K15 

Keywords.

Galois space cap free pair of points fractional factorial design 

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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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