On the size of a cap in \(\text{PG} (n,q)\) with \(q\) even and \(n\geq 3\)
A \(k\)-cap in \(PG(n,q)\) is a set of \(k\) points such that no three of them are collinear. A \(k\)-cap is said to be complete if it is maximal with respect set-theoretic inclusion. The largest value of \(k\) for which a \(k\)-cap in \(PG(n,q)\) exists is denoted my \(m_2(n,q)\) and the size of the second largest complete \(k\)-cap in \(PG(n,q)\) is denoted by \({m'}_2(n,q)\). In the paper under review an improved upper bound to the value \({m'}_2(3,q)\), \(q\) even, \(q\geq 8\) is given. Precisely, the author proves that \[ {m'}_2(3,q)\leq q^2-q+5. \] The new upper bound on \({m'}_2(3,q)\) is then used to improve the upper bound on \({m'}_2(n,q)\), \(q\) even and \(n\geq 4\), namely \[ {m'}_2(n,q)\leq q^{n-1}-q^{n-2}+6q^{n-3}-4q^{n-4}-2(q^{n-5+\ldots +q+1)+1}. \]
scientific article; zbMATH DE number 1270426