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. The effectiveness of the bounds is illustrated by comparison with complete caps sizes obtained by computer searches. The reasonableness of the conjecture is discussed.
Conjectural upper bounds on the smallest size of a complete cap in PG(N, q), N ≥ 3
BARTOLI, DANIELE;FAINA, Giorgio;MARCUGINI, Stefano;PAMBIANCO, Fernanda
2017
Abstract
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. The effectiveness of the bounds is illustrated by comparison with complete caps sizes obtained by computer searches. The reasonableness of the conjecture is discussed.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
