The Constructions Of 1(1/2)-difference Sets

In 1980,Neumaier introduced the concept of t(1/2)-design and gave complete classification of the t(1/2)-design with ? 2.Then Neumaier focused on 112-design.Neumaier researched 1(1/2)-design and gave the relationship of the parameters of 11-design.Then he got some necessary conditions for the existence of 1(1/2)-design and relevant conclusions.The concept of 1(1/2)-design introduced by Neumaier is equivalent to the concept of partial geometric design by Bose.Difference set is one of the classic research questions in combinatorial design theory.It is closely related with design,and it provides good tools and methods for design.In2014,Olmez introduced the concept of 1(1/2)-difference set.In combinatorial design theory,the symmetric 2-designs can be derived from difference sets.Similarly,Olmez pointed out that symmetric 1(1/2)-design can be derived from 112-difference set too.Based on the theory of difference sets,Olmez used group rings and group characters to explore the property of 1(1/2)-difference set.Olmez gave some necessary conditions for the existence of 1(1/2)-difference set,and he obtained some methods to construct 112-difference set.The concept of 1(1/2)-difference set introduced by Olmez is equivalent to the concept of partial geometric difference set by Nowak.Some scholars applied Plateaued functions,perfect nonlinear functions,partial difference set,almost difference set,et.to obtain some conditions for the existence of 1(1/2)-difference set.Based on the conditions for the existence of 1(1/2)-difference sets,some constructions of1(1/2)-difference sets are given in this paper.In addition,we give the existence spectrum of1(1/2)-difference sets with block size 3,4 or 5.