Conjectural upper bounds on the smallest size of a complete cap in PG(N,q), N ≥ 3

In this work we summarize some recent results to be included in a forthcoming paper [Bartoli, D., A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, Upper bounds on the smallest size of a complete cap in PG(N,q) under a certain probabilistic conjecture, preprint]. 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 at every step is considered. It is proved that uncovered points are evenly placed in the space. A natural conjecture on an estimate of the number of new covered points at every step is done. For a part of the iterative process, this estimate is proved rigorously. Under the mentioned conjecture, new upper bounds on the smallest size t2(N,q) of a complete cap in PG(N,q) are obtained. In particular,t2(N,q)<1q−1qN+1(N+1)ln⁡q+1q−3qN+1∼qN−12(N+1)ln⁡q. The effectiveness of the bounds is illustrated by comparison with complete caps sizes obtained by computer searches. The reasonableness of the conjecture is discussed.