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 m 2 + (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 m 2 + (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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