Caps in \(\mathrm{PG}(n,q)\), \(q\) even, \(n\ge 3\)
A \(k\)-cap in \(\mathrm{PG}(n,q)\) is a set of \(k\) points, no three of which are collinear. A \(k\)-cap is called complete if it is not contained in a \((k+1)\)-cap. Let \(m_2(n,q)\) denote the maximum value of \(k\) for which there exists a (complete) \(k\)-cap in \(\mathrm{PG}(n,q)\), and let \(m_2'(n,q)\) denote the second largest value of \(k\) for which there exists a complete \(k\)-cap in \(\mathrm{PG}(n,q)\). The exact value of \(m_2(n,q)\) is only known when \((n,q)=(n,2)\), \((2,q)\), \((3,q)\), (4,3) and (5,3). Much less is known about \(m_2'(n,q)\). A result of \textit{J. W. P. Hirschfeld} and \textit{J. A. Thas} [Geom. Dedicata 23, 15--31 (1987; Zbl 0616.51008)] states that for \(q>2\) an even prime power, \(m_2'(3,q)\le q^2-q/2-\sqrt q/2+2\). By clever counting involving tangent lines the authors are able to show there does not exist a complete \(k\)-cap in \(\mathrm{PG}(3,q)\), \(q\ge 64\) an even prime power, for values of \(k\) in various intervals. Hence they are able to deduce that \(m_2'(3,q)<q^2-q+2\sqrt q+1\). From this result they are then able to show that \(m_2(4,q)<q^ 3-q^2+2q\sqrt q+2q-2\sqrt q+1\) whenever \(q\ge 64\) is an even prime power. An improved upper bound for \(m_2(n,q)\), \(n\ge 5\) and \(q\ge 64\) an even prime power, is also obtained.
scientific article; zbMATH DE number 166876
scientific article