The Arcs In Finite Spaces

This paper concerns the properties of arc in finite projective spaces, and some new theorems are shown and improved. The main result of this paper is a new lower bound of complete k ? arc.Before giving our new results, in Chapter 1, we introduce some background and the connection between k ? arcand maximum distance separable codes. In order to our research of arcs, in Chapter 2, we introduce some theory about the projective space and the properties of arcs. Our main research of this paper is a new lower bound of complete k ? arcin Chapter 4.We know that the correspondence is one to one between M.D. Sand k-arc. So we transform the study of coding theory into the k-arc of limited space. There are some researches about the upper bound of complete k ? arc, and there are many good conclusions. But the research of the lower bound of complete k - arc is not a lot of. In this paper, our main research is on the basis of the known results. We give a new lower bound of complete k-arc, which is k~2 (k-1)>(3/2)q~2-(q+2) if q is even and k~2 (k-1)>(3/2)(q-1)~2-2qk if q is odd.