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

In this work we summarize some recent results to be included in a forthcoming paper [Bartoli, D., A. A. Davydov, A. A. Kreshchuk, S. Marcugini and F. Pambianco, Small complete caps in PG(3,q) and PG(4,q), preprint]. We present and analyze computational results concerning small complete caps in the projective spaces PG(N,q) of dimension N=3 and N=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 new complete caps are the smallest known. Based on them, we obtained new upper bounds on the minimum size t2(N,q) of a complete cap in PG(N,q), N=3,4. Our investigations and results allow to conjecture that these bounds hold for all q.