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Let m_2(n, q) be the maximum size of k for which there exists a k-cap inPG(n, q), and let m'_2(n, q) be the second largest value of k for which thereexists 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 newbounds for m_2(n, q), q even, q\geq 8 and n \geq 4, are obtained. Cao and Oupublished a better bound but there seems to be a gap in their proof.

On k-caps in PG(n, q), with q even and n \geq 3