Upper bounds on the smallest size of a complete cap in $\mathrm{PG}(N, q)$, $N\ge3$, under a certain probabilistic conjecture

In the projective space PG(N, q) over the Galois field of order q, N ≥ 3, an iterative step-by-step construction of complete caps by adding a new point on every step is considered. It is proved that uncovered points are evenly placed on the space. A natural conjecture on an estimate of the number of new covered points on every step is done. For a part of the iterative process, this estimate is proved rigorously. Under the conjecture mentioned, new upper bounds on the smallest size t2(N, q) of a complete cap in PG(N, q) are obtained, in particular,