The Constructions Of 1(1/2)-Design

The t-design is one of the classic problems of combinatorial design.In 1980,Neumaier generalized the concept of t-design to the definition of t(1/2)-design.For t?2,Neumaier gave a complete classification of t(1/2)-design.Each 21/2-design is either a 3-design or a multiple of symmetric 2-design.Each 31/2-design is either a 4-design or a multiple of Hadamard 3-design.When t?4,each t(1/2)-design is a(t+1)-design.Thus,the study of t(1/2)-design is converted to 1(1/2)-design.The concept of 1(1/2)-design given by Neumaier is equivalent to the geometric design given by Bose,Shrikhande,Singhi in 1976.In 1988,Daval generalized the definition of strong regular graph to the concept of directed strongly regular graph.He gave the necessary conditions for the existence of some directed strongly regular graphs and gave a series of construction methods for directed strongly regular graph.However,1(1/2)-design can be used to construct directed strong regular graph.From this point of view,the interest of people' d research on 1(1/2)-design has been set off.In this paper,we forcus on the 1(1/2)-design with fixed block length.We give some constructions of 1(1/2)-design.The existence of the 1(1/2)-design with the length of 4 and ?=1 is completely solved.The existences and non existences of 1(1/2)-designs for some specific parameters are also given.