Upper bounds on the smallest size of a complete cap in $\mathrm{PG}(N,q)$, $N\ge3$, under a certain probabilistic conjecture
Alexander A. Davydov, Giorgio Faina, Stefano Marcugini, Fernanda Pambianco · arXiv (Cornell University) · 2017
In the projective space $\mathrm{PG}(N,q)$ over the Galois field of order $q$, $N\ge3$, 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 $t_{2}(N,q)$ of a complete cap in $\mathrm{PG}(N,q)$ are obtained, in particular, \begin{align*} t_{2}(N,q)