CAP On The Finite Projective Space Research

Coding theory has been a very active field of recent research with a wide range of applications and the closely associated Projective space has also attracted many scholars to study. The first part of the chapter briefly introduces the relevant definitions and finite projective space. The latter part of this chapter is the focus of the work. We mainly improve the upper bound of m2’(3,q) and give the proof of an important theorem.Research on finite projective space of cap has been very popular all over the world. However, it is very difficult to improve its upper bound. So far there are only two precise values. When n=2and q is odd, m2(2,q)=q+1. When n=2and q is even, m2(2,q)=q+2. When n=3, m2(3,q)=q2+1. In order to estimate the upper bound of m2(4,q) and m2(n,q), the method used internationally is to estimate the upper bound of m2’(3,q). Therefore, it is significant to estimate the upper bound of m2’(3,q). For coding theory, each time we obtain a more precise boundary, that is, we will more precisely to find out the length of the linear code.This paper studies the upper bound of the number of elements of caps on the finite projective space. Firstly we use a simper way to prove the conclusion of theorem3.1, which has been proved. On the basis of theorem3.1, we improve the upper bound of m2’(3,q) to q2-3q+16. That is theorem3.2:Let K be the one of complete k-cap of PG(3,q), where q is even and k<q2+1, then k≤q2-3q+16(q≥16). Both theorem3.1and theorem3.2are proved by three sections.The innovation of this paper is to use a simpler method to prove the conclusion of a known theorem. And on this basis, we have reached a new exact value, that is to say we get a smaller upper bound of m2’(3,q).