  Let m_2(n, q) be the maximum size of k for which there exists a k-cap in
PG(n, q), and let m'_2(n, q) be the second largest value of k for which there
exists a complete k-cap in PG(n, q). In this paper Chao's upper bound q^2 - q +
5 for m'_2(3, q), q even and q \geq 8, will be improved. As a corollary new
bounds for m_2(n, q), q even, q\geq 8 and n \geq 4, are obtained. Cao and Ou
published a better bound but there seems to be a gap in their proof.
