New upper bounds on the smallest size of a complete cap in the spaces PG(3, q) and PG(4, q)

In this paper we present and analyze computational results concerning small complete caps in the projective spaces PG(N, q) of dimension N = 3, 4 over the finite field of order q. The results have been obtained using randomized greedy algorithms and the algorithm with fixed order of points (FOP). The computatiohs have been done in wide regions of q values; such wide regions are not considered in literature for N = 3, 4. The new complete caps are the smallest known. Basing on these, we obtained new upper bounds on t(2)(N, q), the smallest size of a complete cap in PG(N, q), in particular,t2(N, q) < root N + 2 . q(N - 1/2) root ln q, q is an element of L-N, N = 3, 4;t(2)(N, q) < (root N + 1 + (1.3) (ln(2q))) q(N - 1/2) root ln q, q is an element of L-N, N = 3,4.Here L-N is a region of q values in which the computations were done. Moreover, our investigations and results allow to conjecture that these bounds hold for all q.