Recursive constructions of complete caps (Q5935436)

A cap \(K\) is a set of points in \(PG(d,q)\) no three of which are collinear. A cap is complete if it is not contained in any larger cap. The minimum number of points in a complete cap is denoted by \(n_2(d,q)\). Little is known about small complete caps for large dimensions. Upper bounds for \(n_2(d,q)\) are known for \(q\) even, but there is little in the literature concerning the odd characteristic case. NEWLINENEWLINENEWLINEIn this paper the authors present three recursive constructions of complete caps when \(q\) is odd. These constructions imply the following results:NEWLINENEWLINENEWLINEFor \(k \geq 0\), NEWLINE\[NEWLINE n_2(k+1,3) \leq 2n_2(k,3).NEWLINE\]NEWLINE Let \(q \geq 5\) be an odd prime power and \(k \geq 0\). Then NEWLINE\[NEWLINEn_2(4k+2,q) \leq q^{2k+1} + n_2(2k,q).NEWLINE\]NEWLINE And, let \(q \geq 9\) be an odd prime power and \(k \geq 0\). Then NEWLINE\[NEWLINEn_2(4k+2,q) \leq q^{2k+1} - (q+1) + n_2(2k,q) + n_2(2,q).NEWLINE\]