Partial Geometric Difference Sets And Related Designs

Bose,Shrikhande and Singhi first proposed the concept of partial geometric design in 1976.Subsequently,Bose et al.extensively studied the algebra and combinatorial properties of partial geometric designs.In 2012,Brouwer et al.pointed out that two classes of directed strongly regular graphs could be constructed by using flags and anti-flags of partial geometric designs,which triggered a new research boom in partial geometric design.Difference set is one of the classical problems in combinatorial design.As a generalization of the difference set,the partial geometric difference set is closely related to the partial geometric design and the divisible difference set.Olmez first proposed the concept of 1 1/2-difference set in 2014,which is the partial geometric difference set to be studied in this paper.Similar to the method that the symmetric 2-design can be derived from the difference set,Olmez obtained the conclusion that the partial geometric difference set can be used to construct the symmetric partial geometric design.Subsequently,many scholars have given some construction methods using various differential sets and algebraic structures such as μ-palteaued functions.In this paper,some constructions of partial geometric difference sets are given,some new partial geometric difference sets are obtained,by using the combinatorial structures such as the divisible difference sets and the theories such as group rings and direct products,and then some new partial geometric designs and directed strongly regular graphs are obtained.