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 [2]. 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) < 1 q − 1 √ qN+1(N + 1) ln q + 1 q − 3 √ qN+1 ∼ q N−1 2 √ (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.