1(1/2)-Diffrence Family And Its Constructions

In 1976, Bose et. al. puts forward the concept of 1(1/2)sign or part of geometric design, Neumaier generalized the conception to t(1/2)design in 1980. And he gave the complete classification of t(1/2)design for t≥ 2. Then, the scholars gave much attention to 1(1/2)designs. Brouwer et. al. found the constructions of some directed strongly regular graphs by using the flags or anti-flags of 1(1/2)designs as points. Since then,1(1/2)-designs became a significant concern.The difference sets are the important research objects in combinatorial design the-ory. Olmez introduced the concept of 1(1/2)difference set by using group ring and group characters. He investigated the necessary conditions of 1(1/2)-difference sets and provided some nonexistence results of 1(1/2)designs. In addition, Olmez showed that some symmetric 1(1/2)designs can be constructed from 1(1/2)difference sets.The difference family is the generalization of the difference set. Similarly, we gener-alized the 1(1/2)difference sets to the 1(1/2)difference family. In this paper, we investigate the relationship between 1(1/2)difference family and 1(1/2)design. Some existence of 1(1/2)difference families with block size 3 or 4 are obtained by calculating the occurrences of differences. Some general constructions of 1(1/2)fference families are given.