New inductive constructions of complete caps in<i>PG</i>(<i>N, q</i>),<i>q</i>even

Abstract Some new families of small complete caps in PG ( N, q ), q even, are described. By using inductive arguments, the problem of the construction of small complete caps in projective spaces of arbitrary dimensions is reduced to the same problem in the plane. The caps constructed in this article provide an improvement on the currently known upper bounds on the size of the smallest complete cap in PG ( N, q ), N ≥4, for all q ≥2 3 . In particular, substantial improvements are obtained for infinite values of q square, including q =2 2 Cm , C ≥5, m ≥3; for q =2 Cm , C ≥5, m ≥9, with C , m odd; and for all q ≤2 18 . © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 177–201, 2010