Small complete caps inPG(N,q),qeven

Abstract

AbstractA new family of small complete caps inPG(N,q),qeven, is constructed. Apart from small values of eitherNorq, it provides an improvement on the currently known upper bounds on the size of the smallest complete cap inPG(N,q): forNeven, the leading term$q^{{N}\over{2}}$is replaced by$\alpha q^{{N}\over{2}}$with$\alpha \le {{1}\over{2}}$, forNodd the leading term$3q^{{N-1}\over{2}}$is replaced by$\beta q^{{N-1}\over{2}}$with$\beta \le {{5}\over{2}}$. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 420–436, 2007