A New Lower Bound Of The Complete K-arcs

This paper concern the theorems of arcs in finite projective spaces and a new lower bound of complete k-arc.New proofs to some theorems are shown in this paper. We get a series of new findings which partly improve or extend the related results in the literatures. Our main results of this paper are in the chapter 3 and chapter 4 , and what we mostly focus on is a new lower bound of complete k-arc K in PG(n,q).Before showing our results, we give a brief introduction about the background of our research as well as its applications in coding theory in the chapter 1. At the same time, we introduce many concepts and useful properties about the projective space in the chapter 2, which are intimately related to our study.Our main research is in the chaper 3. In this chapter, we give a new lower bound of complete k-arc K, which is k~3 - k~2 > ( q + 2)~2- k ( q+ 2) if q is even and at the same time if q is odd k~3 - k~2 > ( q + 2)~2- k ( q+ 2). It is a better lower bound of complete k-arc. In the beginning of this chapter, we introduce definition of k-arcs and the theorems of complete k-arcs and most results about complete k-arc are improve by a new elem- entary method.Finally, we calculate and summarize the q-multinomial coefficients of combina- torial mathematics. And we connect them to the projective space.