Constructions Of Partial Geometric Difference Sets And Partial Geometric Difference Families

The notion of a partial geometric design was first introduced by Bose,Shrikhande,and Singhi[7]in 1976.Subsequently,Bose et al.[4-6]widely investigated algebraic and combinatorial properties of partial geometric designs.In the process of study of t(1/2)?designs in 1980,Neumaier1271 called partial geometric designs 1(1/2)-designs as a sub-class.In 2005,Van Dam and Spencec[12]studied partial geometric designs as a type of combinatorial designs whose incidence matrix has two distinct singular values,and constructed several partial geometric designs with small parameters.Chai et al.[9]con-structed partial geometric designs from symplectic geometry over infinite fields.Duval[14]defined the directed strongly regular graphs in 1988 as a generalization of strongly regular graph.Since there are many limitations on the possible sets of its parameters,directed strongly regular graphs are fairly rare.However,there are many constructions methods,for example,combinatorial designs[17],coherent algebras[22],finite geometries[16],regular tournaments[21],block matrices[1]and Cayley graphs[15]and so on.In 2012,Brouwer et al.[8]showed that two directed strongly regular graphs can be constructed from a given partial geometric design.One is defined on the set of flags of the design and the other is defined on the set of antiflags.From this point of view,the construction of directed strongly regular graphs can be converted to the construction of partial geometric designs.In 2013,Olmez[34]introduced the concept of a partial geometric difference set,under the name of "1(1/2)-difference set",as a difference set version of partial geometric design and he demonstrated that we can obtain symmetric partial geometric designs from partial geometric difference sets in precisely the same manner that we obtain sym-metric 2-designs from ordinary difference sets.Further,Nowak et al.[31]introduced the notion of a partial geometric difference family,which generalizes both the classical difference family and the partial geometric difference set.It was shown that a partial geometric difference family also gives rise to a partial geometric design.Very recently,Michel[26]constructed several new infinite classes of partial geometric difference sets and partial geometric difference families,mostly by similar approaches of Nowak et al.[30,31],but also from planar functions.This paper mainly focuses on general constructions for partial geometric difference sets and partial geometric difference families,and is organized as follows.Chapter 1 recalls some preliminary notations and terminologies that will be used throughout the paper.In Chapter 2,we display a few general constructions of partial geometric differ-ence sets in the direct product of two groups,and with applications we get a number of new families of partial geometric sets.Our constructions unify and generalize a few known results of Michell[26],Olmez[33]and Spence[36].In Chapter 3 we pro-duce several new families of partial geometric difference families.Finally,in Chapter 4,we summarize the new parameters of the partial geometric designs obtained from our previous constructions,thereby giving directed strongly regular graphs with new parameters.