No abstract

We introduce the notion of a partial geometric difference family as a variation on the classical difference family and a generalization of partial geometric difference sets. We study the relationship between partial geometric difference families and both partial geometric designs and difference families, and show that partial geometric difference families give rise to partial geometric designs. We construct several infinite families of partial geometric difference families using Galois rings and the cyclotomy of Galois fields. From these partial geometric difference families, we generate a list of infinite families of partial geometric designs and directed strongly regular graphs.

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.

In this paper, we show that partial geometric designs can be constructed from certain three‐class association schemes and ternary linear codes with dual distance three. In particular, we obtain a family of partial geometric designs from the three‐class association schemes introduced by Kageyama, Saha, and Das in their article [“Reduction of the number of associate classes of hypercubic association schemes,” Ann Inst Statist Math 30 (1978)]. We also give a list of directed strongly regular graphs arising from the partial geometric designs obtained in this paper.

Let G be an additive group of order v. A k-element subset D of G is called a (v, k, λ, t)-almost difference set if the expressions gh −1 , for g and h in D, represent t of the non-identity elements in G exactly λ times and every other non-identity element λ + 1 times. Almost difference sets are highly sought after as they can be used to produce functions with optimal nonlinearity, cyclic codes, and sequences with three-level autocorrelation. This paper reviews the recent work that has been done on almost difference sets and related topics. In this survey, we try to communicate the known existence and nonexistence results concerning almost difference sets. Further, we establish the link between certain almost difference sets and binary sequences with three-level autocorrelation. Lastly, we provide a thorough treatment of the tools currently being used to solve this problem. In particular, we review many of the construction methods being used to date, providing illustrative proofs and many examples.