An Elementary Proof Of B.Segre Theorem And A Bound Of T-blocking Sets

The maximum distance separating code(M.D.S code) is the code that can correctthe most errors in coding theory,and the correspondence between the code and (k, r) arcs is one to one relation.So,we can transform the study of M.D.S code into the (k, r) arcs of finite projective space.In this thesis,the priority we study are the lower boundof t blocking sets and the upper bound of complete k arcs in P G(2, q).First,we havestudied the lower bound of t blocking sets in P G(2, q),and proved a new lower boundof t blocking sets.Then we study the character of k arcs,and give an elementaryproof of B.Segre Theorem which gives an important upper bound of k arcs.We havemade some improvements to the conclusion.Before showing the main results,we give a brief introduction about back ground ofour research as well as its applications in coding theory.At the same time, we introducemany concepts and useful properties about the projective space,and this is the maincontent of the2nd chapter,also the basement part we study.The main results we studyare mentioned in the3rd and4th chapters.The first main result is that we proved a new lower bound of t blocking setsin P G(2, q).In fact,the new lower bound is the improvement of Ball Theorem.We usetwo diferent methods to get the new bound.This method we use is always used in thisthesis. The2rd main result is the improvement of the upper bound of k arcs.First,wegive B.Segre Theorem an elementary proof with our method,and then make some im-provement to it,and then we get some better new upper bounds.