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

In 1980,Neumaier first proposed the concept of t(1/2)-designs,and completely classified t(1/2)-designs with t≥2.Therefore,the research on t(1/2)-designs was transformed into the research on 1(1/2)-designs.After that,many mathematicians are devoted to the research on 1(1/2)-designs.The concept of partial geometric designs which proposed by Bose in 1976 is equivalent with 1(1/2)-designs proposed by Neumaier in 1976.In 2014,Olmez introduced the definition of 1(1/2)-difference sets.He proved that 1(1/2)-difference sets can obtain 1(1/2)-designs.He studied the existence and nonexistence of 1(1/2)-difference sets,and gave some constructions of 1(1/2)-difference sets.In 2016,Meng gave the concept of 1(1/2)-difference families and the necessary conditions for its existence.She con-structed some new 1(1/2)-difference families.In the same year,Nowak introduced the concept of geometric difference families(i.e.1(1/2)-difference families)and studied the necessary con-ditions for the existence of 1(1/2)-difference families.Some new 1(1/2)-difference families were constructed.In 2018,Chang etc promoted the constructions of 1(1/2)-difference sets,and obtained some more general constructions of 1(1/2)-difference sets.At the same time,they also con-structed some new 11/1-difference families.In the same year,Shen and Zhang respectively constructed some new 11/1-difference sets and 1(1/2)-difference families in their thesis,and discussed the existence of some 1(1/2)-difference sets and 1(1/2)-difference families.In this paper,according to the combinatorial properties of 1(1/2)-difference sets,some new 1(1/2)-difference sets are constructed,and the existence of 1(1/2)-difference families with block size four is further discussed.