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

The 1(1/2)-design(or partial geometric design)was first proposed by Bose et al.in 1976.In 1980,Neumaier discussed 1(1/2)-designs in depth during the study of t(1/2)-designs.In recent years,Brouwer et al.proved that directed strongly regular graphs can be constructed by using flags or anti-flags of 1(1/2)-designs.Since then,mathematicians have turned the focus of research into 1(1/2)-designs.In 2016,Meng introduced the definition of 1(1/2)-difference family and gave some prerequisites for the existence of 1(1/2)-difference families.She proposed some methods for constructing 1(1/2)-difference families.In addition,she discussed 1(1/2)-difference families with block size 3 and 4.In the same year,Nowak et al.put forward the concept of partial geometry difference family,which is equivalent to the case where the 1(1/2)-difference family does not contain short orbits.Furthermore,they analyzed the properties of 1(1/2)-difference families and constructed 1(1/2)-difference families over finite fields and Galois rings.Nowak and Olmez constructed 1(1/2)-difference families using hyperplanes of vector spaces.The1(1/2)-difference family is the extension of the 1(1/2)-difference set and the difference family.During recent years,the 1(1/2)-difference family has attracted greater attention because it is one of the important tools to construct 1(1/2)-designs.The goal of this paper is to study the 1(1/2)-difference families without short orbits.Based on the existing methods of constructing 1(1/2)-difference sets and 1(1/2)-difference families and algebraic tools such as group ring,we give some new recursive constructions and directly constructions of 1(1/2)-difference families.In addition,we study the relations between the parameters and the structures of 1(1/2)-difference families with block size 4 and discuss their existence.