On k-caps in PG(n, q), with q even and n \geq 4

Let $m_2(n,q),n\ge 3$, be the maximum size of k for which there exists a complete k-cap in PG(n, q). In this paper the known bounds for $m_2(n, q), n \geq 4$,qevenand$q \geq 2048$, will be considerably improved.