arXiv:1610.09656v1 [math.CO] 30 Oct 2016 Tables, bounds and graphics of the smallest known sizes of complete caps in the spaces PG(3,q ) and PG(4,q ) ∗ Daniele Bartoli Dipartimento di Matematica e Informatica, Universit` a degli Studi di Perugia, Via Vanvitelli 1, Perugia, 06123, Italy. E-mail: daniele.bartoli@unipg.it Alexander A. Davydov, Alexey A. Kreshchuk Institute for Information Transmission Problems (Kharkevich institute) Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, GSP-4, Moscow, 127994 Russian Federation. E-mail: {adav,krsch}@iitp.ru Stefano Marcugini and Fernanda Pambianco Dipartimento di Matematica e Informatica, Universit` a degli Studi di Perugia, Via Vanvitelli 1, Perugia, 06123, Italy. E-mail: {stefano.marcugini,fernanda.pambianco}@unipg.it Abstract In this paper 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 computations have been done in relatively 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 them, we obtained new upper bounds on t 2 (N,q), the minimum size of a complete cap in PG(N,q), in particular, t 2 (N,q) < √ N +2 · q N-1 2  ln q, q ∈ L N , N =3, 4, * The research of D. Bartoli, S. Marcugini and F. Pambianco was supported in part by Ministry for Education, University and Research of Italy (MIUR) (Project “Geometrie di Galois e strutture di inci- denza”) and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INDAM). The research of A.A. Davydov and A.A. Kreshchuk was carried out at the IITP RAS at the expense of the Russian Foundation for Sciences (project 14-50-00150). 1