The New Bound Of T-Blocking And Cap

In this paper, we are concerned with the upper bound of t-blocking sets and caps in finite dimensional projective spaces. We not only improve some given results, but also prove 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 sets in the PG (2,q). We have improved the Bruen-Thas theorem, which is about the upper bound of t-blocking sets. In the chapter 4, the upper bound of the size of caps is our main research, namely, a new upper bound of m₂(n,q) in PG (n,q). To estimate m₂(n,q), we firstly estimate m₂(4,q ) when n=4, then induce conclusion in the PG (n,q). The method in common use is to estimate m₂'(3,q) in the first step, and then get a new estimated value of m₂(4,q) by the m₂'(3,q) . We calculate the upper bound of m₂(4,q) directly with more elementary method. In the beginning of this chapter, most results about m₂(n,q) and m₂'(n,q) are shown.Before giving our results, in chapter 1, we introduce some background and applications of our research.In chapter 2, we introduce many concepts and natures about the projective space.