| In this paper, we are concerned with the sizes of t-blocking sets, (k, r)-arcs and caps in finite projective spaces, not only giving one theorem an elementary proof, improving some theorem' s results, but also proving some new exact values. The main results of this paper are in the chapter 3 and chapter 4, and what we mostly focus on is the caps of the chapter 4.In the chapter 3, we study the t-blocking set and (k, r)-arc in the PG (2, q). First we give Ball' s theorem an elementary proof and improve its result, then we prove a new value of mr(2, q). In the chapter 4, the upper bound of thecaps is our main research. Firstly, we prove a better upper bound of m'2 (3, q) when q is even and q= 2e, e≥8. And then, we get a better upper bound of m2(4, q) based on the former result. Lastly, we induce our conclusion inthe PG (n, q) and receive a better upper bound of m2(n, q). In the beginningof this chapter, most results about m2(n, q) and m'2 (n, q) are shown.Before giving our results, in chapter 1, we introduce some background and application fields of our research. In chapter 2, we introduce many concepts and natures about the projective space. |
