On the size of complete caps in PG\((3,2^{h}\))
A \(k\)-cap in \(\text{ PG}(n,q)\) 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 \(m_2(n,q)\) (respectively \(m_2'(n,q)\)) denote the largest (respectively second largest) value of \(k\) for which there exists a complete \(k\)-cap in \(\text{ PG}(n,q)\). It is well known that \(m_2(3,q)=q^2+1\) if \(q \not= 2\). By [\textit{J. M. Chao}, Geom. Dedicata 74, No. 1, 91--94, (1999; Zbl 0920.51004)], we know that \(m_2'(3,q) \leq q^2-q+5\) if \(q\) is an even prime power which is at least 8. In the present paper the authors improve this bound. They show that \(m_2'(3,q) \leq q^2-q+2\) if \(q\) is an even prime power which is at least 16. They also improve a number of intervals for \(k\) for which there does not exist a complete \(k\)-cap in \(\text{ PG}(3,q)\), \(q\) even.
scientific article; zbMATH DE number 2085841
scientific article