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Partial geometric difference sets (PGDSs) were defined in Olmez (J Combin Des 22(6):252-269, 2014). They are used to construct partial geometric designs. We use the framework of extended building sets to find infinite families of PGDSs in abelian groups. Included in our new families of PGDSs are generalizations of the Hadamard, McFarland, Spence, Davis-Jedwab, and Chen difference sets. [ABSTRACT FROM AUTHOR]

In this paper, we show that partial geometric designs can be constructed from certain three-weight linear codes, almost bent functions and ternary weakly regular bent functions. In particular, we show that existence of a family of partial geometric difference sets is equivalent to existence of a certain family of three-weight linear codes. We also provide a link between ternary weakly regular bent functions, three-weight linear codes and partial geometric difference sets. [ABSTRACT FROM AUTHOR]

Combinatorial designs have long been used to design efficient statistical experiments. More recently, connections to the theory of cryptographic communications have emerged. Combinatorial designs have provided solutions to problems coming from signal processing, radar, error-correcting codes, optical orthogonal codes, and image processing. Further, the most elegant solutions have come from designs with prescribed automorphisms. In this paper, we focus on partial geometric designs, a generalization of the classical 2-design. These designs have recently been shown to produce two directed strongly regular graphs. We generalize the well-known Kramer-Mesner Theorem for 2-designs to partial geometric designs. We also construct infinite families of partial geometric designs admitting a group of automorphisms which acts regularly on the point set and semi-regularly on the block set. The designs are obtained from new constructions of partial geometric difference families. These families were recently introduced a generalization of both the classical difference family and the partial geometric difference set. [ABSTRACT FROM AUTHOR]

We construct an infinite family of $$1\frac{1}{2}$$ -difference sets in non-cyclic abelian $$p$$ -groups. In particular, we examine the construction in $$2$$ -groups to discover the useful relationship between $$1\frac{1}{2}$$ -difference sets and certain Boolean functions. [ABSTRACT FROM AUTHOR]

We present many new directed strongly regular graphs by direct construction. We construct these graphs on the collections of antiflags of certain finite incidence structures. In this way, we confirm the feasibility of infinitely many parameter sets that was previously undetermined. We describe some examples of graphs together with their isomorphism classes to demonstrate the fact that our construction method is capable of producing many graphs with same parameters. [ABSTRACT FROM AUTHOR]