Caps in with even and

Let m 2 ( n , q ) , m 2 ′ ( n , q ) be, respectively, the maximum value, the second largest value of k for which there exists a complete k -cap in P G ( n , q ) . In this paper, the known upper bound on m 2 ′ ( 3 , q ) , q even, q ≥ 128 , is improved. This new upper bound on m 2 ′ ( 3 , q ) is then used to improve the upper bounds on m 2 ( n , q ) , q even, q ≥ 128 and n ≥ 4 . However, only when q ≥ 128 , n = 4 and n = 5 , the result of m 2 ( n , q ) is better than the result of S. Ferret and L. Storme.