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

Let m2(n,q) be the maximum size of k for which there exists a k-cap in PG(n,q), and let m2′(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 q2−q+5 for m2′(3,q), q even and q≥8, will be improved. As a corollary new bounds for m2(n,q), q even, q≥8 and n≥4, are obtained. Cao and Ou published a better bound but there seems to be a gap in their proof.