New upper bounds for the sizes of caps in finite projective spaces (Q1610991)

The authors obtain in this article improved upper bounds for the sizes of caps in finite projective spaces. Let \(m_{2}(N,q)\) be the size of a maximal cap in \(PG(N,q)\). After proving that large caps can be uniquely extended to complete ones, they improve the upper bound of \textit{G. P. Nagy} and \textit{T. Szönyi} [J. Geom. 59, 103-113 (1997; Zbl 0884.51009)] for \(m_{2}(4,q)\), \(q\) odd. An improvement of the Hill formula relating \(m_{2}(N,q)\) and \(m_{2}(N-1,q)\) is then obtained, using the cardinalities of the intersection of a cap with two given subspaces of \(PG(N,q)\) and known bounds for \(m_{2}(N,q)\). This ameliorated inductive relation is used to improve the upper bounds on \(m_{2}(N,q)\), \(N>4\), \(q\) odd. The last section is devoted to the case \(q\) even, a new upper bound for \(m_{2}(N,q)\) (\(N>4\)) being found using some arguments developed by \textit{J. W. P. Hirschfeld} and \textit{J. A. Thas} [Geom. Dedicata 23, 15-31 (1987; Zbl 0616.51008), General Galois Geometries (Oxford Mathematical Monographs, Clarendon Press, Oxford) (1991; Zbl 0789.51001)] and without use of Hill's formula.