A ( v , k , λ , μ ) -partial difference set (PDS) is a subset D of a group G such that | G | = v , | D | = k , and every nonidentity element x of G can be written in either λ or μ different ways as a product g h - 1 , depending on whether or not x is in D . Assuming the identity is not in D and D is inverse-closed, the corresponding Cayley graph Cay ( G , D ) will be strongly regular. Partial difference sets have been the subject of significant study, especially in abelian groups, but relatively little is known about PDSs in nonabelian groups. While many techniques useful for abelian groups fail to translate to a nonabelian setting, the purpose of this paper is to show that examples and constructions using abelian groups can be modified to generate several examples in nonabelian groups. In particular, in this paper we use such techniques to construct the first known examples of PDSs in nonabelian groups of order q 2 m , where q is a power of an odd prime p and m ≥ 2 . The groups constructed can have exponent as small as p or as large as p r in a group of order p 2 r . Furthermore, we construct what we believe are the first known Paley-type PDSs in nonabelian groups and what we believe are the first examples of Paley–Hadamard difference sets in nonabelian groups, and, using analogues of product theorems for abelian groups, we obtain several examples of each. We conclude the paper with several possible future research directions.

We construct two families of strongly regular Cayley graphs, or equivalently, partial difference sets, based on elementary abelian groups. The parameters of these two families are generalizations of the Denniston and the dual Denniston parameters, in contrast to the well known Latin square type and negative Latin square type parameters. The two families unify and subsume a number of existing constructions which have been presented in various contexts such as strongly regular graphs, partial difference sets, projective sets, and projective two-weight codes, notably including Denniston's seminal construction concerning maximal arcs in classical projective planes with even order. Our construction generates further momentum in this area, which recently saw exciting progress on the construction of the analogue of the famous Denniston partial difference sets in odd characteristic.

Denniston constructed partial difference sets (PDSs) with the parameters (23m,(2m+r−2m+2r)(2m−1),2m−2r+(2m+r−2m+2r)(2r−2),(2m+r−2m+2r)(2r−1)) in elementary abelian groups of order 23m for all m≥2,1≤r<m. These correspond to maximal arcs in Desarguesian projective planes of even order. In this paper, we show that - although maximal arcs do not exist in Desarguesian projective planes of odd order - PDSs with the Denniston parameters (p3m,(pm+r−pm+pr)(pm−1),pm−pr+(pm+r−pm+pr)(pr−2),(pm+r−pm+pr)(pr−1)) exist in all elementary abelian groups of order p3m for all m≥2,r∈{1,m−1} where p is an odd prime, and present a construction. Our approach uses PDSs formed as unions of cyclotomic classes.

For nearly a century, mathematicians have been developing techniques for constructing abelian automorphism groups of combinatorial objects, and, conversely, constructing combinatorial objects from abelian groups. While abelian groups are a natural place to start, recent computational evidence strongly indicates that the vast majority of transitive automorphism groups of combinatorial objects are nonabelian. This observation is the guiding motivation for this paper. We propose a new method for constructing nonabelian automorphism groups of combinatorial objects, which could be called the combinatorial transfer method, and we demonstrate its power by finding (1) the first infinite families of nonabelian Denniston partial difference sets (including nonabelian Denniston PDSs of odd order), (2) the first infinite family of Spence difference sets in groups with a Sylow 3-subgroup that is non-normal and not elementary abelian, (3) the first infinite families of McFarland difference sets in groups with a Sylow p-subgroup that is non-normal and is not elementary abelian, (4) new infinite families of partial difference sets in nonabelian p-groups with large exponent, (5) an infinite family of semiregular relative difference sets whose forbidden subgroup is nonabelian, and (6) a converse to Dillon's Dihedral Trick in the PDS setting. We hope this paper will lead to more techniques to explore this largely unexplored topic.

Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which-one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil-are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

Spence [9] constructed ( 3^d+1(3^d+1-1)/2, 3^d(3^d+1+1)/2, 3^d(3^d+1)/2) -difference sets in groups K × C_3^d+1 for d any positive integer and K any group of order 3^d+1-1/2 . Smith and Webster [8] have exhaustively studied the d=1 case without requiring that the group have the form listed above and found many constructions. Among these, one intriguing example constructs Spence difference sets in A_4 × C_3 by using (3, 3, 3, 1)-relative difference sets in a non-normal subgroup isomorphic to C_3^2 . Drisko [3] has a note implying that his techniques allow constructions of Spence difference sets in groups with a noncentral normal subgroup isomorphic to C_3^d+1 as long as 3^d+1-1/2 is a prime power. We generalize this result by constructing Spence difference sets in similar families of groups, but we drop the requirement that 3^d+1-1/2 is a prime power. We conjecture that any group of order 3^d+1(3^d+1-1)/2 with a normal subgroup isomorphic to C_3^d+1 will have a Spence difference set (this is analogous to Dillon’s conjecture in 2-groups, and that result was proved in Drisko’s work). Finally, we present the first known example of a Spence difference set in a group where the Sylow 3-subgroup is nonabelian and has exponent bigger than 3. This new construction, found by computing the full automorphism group Aut(𝒟) of a symmetric design associated to a known Spence difference set and identifying a regular subgroup of Aut(𝒟) , uses (3, 3, 3, 1)-relative difference sets to describe the difference set.

Denniston constructed partial difference sets (PDSs) with the parameters $(2^{3m}, (2^{m+r} - 2^m + 2^r)(2^m-1), 2^m-2^r+(2^{m+r}-2^m+2^r)(2^r-2), (2^{m+r}-2^m+2^r)(2^r-1))$ in elementary abelian groups of order $2^{3m}$ for all $m \geq 2, 1 \leq r < m$. In this paper, we show that PDSs with the Denniston parameters $(p^{3m}, (p^{m+r} - p^m + p^r)(p^m-1), p^m-p^r+(p^{m+r}-p^m+p^r)(p^r-2), (p^{m+r}-p^m+p^r)(p^r-1))$ exist in all elementary abelian groups of order $p^{3m}$ for all $m \geq 2, r \in \{1, m-1\}$ where $p$ is an odd prime, and present a construction. Our approach uses PDSs formed as unions of cyclotomic classes.

In this paper, we consider the problem of linking systems of difference sets as well as variations. We review recent results on the subject, give new constructions of linked relative difference sets and relative linking systems of nonreversible difference sets, and conclude with several open questions.

Collections of difference sets called linking systems have been used to construct new families of linked systems of symmetric designs. In this paper, we define relative and almost linking systems, collections of difference sets and almost difference sets with very similar linking properties to linking systems. These linking systems have connections to bent sets and vectorial bent functions. We construct examples of relative and almost linking systems using a technical lemma.

In this paper we construct several infinite families of partial difference sets of both the Latin and negative Latin square type. Among these constructions is a new family having parameters \((3^{2t},r(3^t+1),-n+r^2+3r,r^2+r)\), where \(r=3^{t-1}+1\) (new for \(t \ge 4\)). For the cases where \(r = 3^{t-1}-1\) and \(3^{t-1}\), the constructions generalize previous results to a larger collection of abelian groups.

Linked systems of symmetric designs are equivalent to 3-class Q-antipodal association schemes. Only one infinite family of examples is known, and this family has interesting origins and is connected to important applications. In this paper, we define linking systems, collections of difference sets that correspond to systems of linked designs, and we construct linking systems in a variety of nonelementary abelian groups using Galois rings, partial difference sets, and a product construction. We include some partial results in the final section.

We present new abelian partial difference sets and amorphic group schemes of both Latin square type and negative Latin square type in certain abelian p-groups. Our method is to construct what we call pseudo-quadratic bent functions and use them in place of quadratic forms. We also discuss a connection between strongly regular bent functions and amorphic group schemes.

Relatively few constructions are known of negative Latin square type Partial Difference Sets (PDSs), and most of the known constructions are in elementary abelian groups. We present a product construction that produces negative Latin square type PDSs, and we apply this product construction to generate examples in p-groups of exponent bigger than p.

In this paper we give some necessary and sufficient conditions for Dembowski-Ostrom polynomials to be planar. These conditions give a simple explanation of the Coulter-Matthews and Ding-Yin commutative semifields and enable us to obtain permutation polynomials from some of the Zha-Kyureghyan-Wang commutative semifields. We then give a generalization of Feng's construction of Paley type group schemes in extra-special p-groups of exponent p and construct a family of Paley type group schemes in what we call the flag groups of finite fields. We also determine the strong multiplier groups of these group schemes. In the last section of this paper, we give a straightforward generalization of the twin prime power construction of difference sets to a construction of Hadamard designs from twin Paley type association schemes.

Doubly Regular Asymmetric Digraphs (DRAD) with rank 4 automorphism groups were previously thought to be rare. We exhibit difference sets in Galois Rings that can be used to construct an infinite family of DRADs with rank 4 automorphism groups. In addition, we construct difference sets in groups Z2r2 for all r⩾2 that can be used to construct DRADs and nonsymmetric 3-class imprimitive association schemes. Finally, we prove a new product construction for difference sets so that the resulting difference sets can be used to build nonsymmetric 3-class imprimitive association schemes.

A partial difference set with parameters (v,v−12,v−54,v−14) is said to be of Paley type. In this paper, we give a recursive theorem that for all odd n>1 constructs Paley partial difference sets in certain groups of order n4 and 9n4. We are also able to construct Paley–Hadamard difference sets of the Stanton–Sprott family in groups of order n4(n4±2) when n4±2 is a prime power and 9n4(9n4±2) when 9n4±2 is a prime power. Many of these are new parameters for such difference sets, and also give new Hadamard designs and matrices.

By modifying a construction for Hadamard (Menon) difference sets we construct two infinite families of negative Latin square type partial difference sets in groups of the form \({\mathbb {Z}_3^2 \times \mathbb {Z}_p^{4t}}\) where p is any odd prime. One of these families has the well-known Paley parameters, which had previously only been constructed in p-groups. This provides new constructions of Hadamard matrices and implies the existence of many new strongly regular graphs including some that are conference graphs. As a corollary, we are able to construct Paley–Hadamard difference sets of the Stanton-Sprott family in groups of the form \({\mathbb {Z}_3^2 \times \mathbb {Z}_p^{4t} \times {\it EA} (9p^{4t} \pm 2)}\) when \({9p^{4t} \pm 2}\) is a prime power. These are new parameters for such difference sets.

A partial difference set (PDS) having parameters (n(2), r(n-1), n+r(2)-3r, r(2)-r) is called a Latin square type PDS, while a PDS having parameters (n(2), r(n+1), n+r(2)+3r, r(2)-r) is called a negative Latin square type PDS. There are relatively few known constructions of negative Latin square type PDSs, and nearly all of these are in elementary abelian groups. We show that there are three different groups of order 256 that have all possible negative Latin square type parameters. We then give generalized constructions of negative Latin square type PDSs in 2-groups. We conclude by discussing how these results fit into the context of amorphic association schemes and by stating some open problems. (C) 2009 Wiley Periodicals, Inc. J Combin Designs 17: 266-282, 2009

A partial difference set having parameters (n 2, r(n − 1), n + r 2 − 3r, r 2 − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n 2, r(n + 1), − n + r 2 + 3r, r 2 + r) is called a negative Latin square type partial difference set. Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups. In this paper, we develop three product theorems that construct negative Latin square type partial difference sets and Latin square type partial difference sets in direct products of abelian groups G and G′ when these groups have certain Latin square or negative Latin square type partial difference sets. Using these product theorems, we can construct negative Latin square type partial difference sets in groups of the form \({G = (Z_2)^{4s_0} \times (Z_4)^{2s_1} \times (Z_{16})^{4s_2} \times \cdots \times (Z_{2^{2r}})^{4s_r}}\) where the s i are nonnegative integers and s 0 + s 1 ≥ 1. Another significant corollary to these theorems are constructions of two infinite families of negative Latin square type partial difference sets in 3-groups of the form \({G = (Z_3)^2 \times (Z_3)^{2{s_1}} \times (Z_9)^{2{s_2}} \times \cdots \times (Z_{3^{2k}})^{2{s_k}}}\) for nonnegative integers s i . Several constructions of Latin square type PDSs are also given in p-groups for all primes p. We will then briefly indicate how some of these results relate to amorphic association schemes. In particular, we construct amorphic association schemes with 4 classes using negative Latin square type graphs in many nonelementary abelian 2-groups; we also use negative Latin square type graphs whose underlying sets can be elementary abelian 3-groups or nonelementary abelian 3-groups to form 3-class amorphic association schemes.

A partial difference set having parameters $(n^2, r(n-1), n+r^2-3r,r^2-r)$ is called a Latin square type partial difference set, while a partial difference set having parameters $(n^2, r(n+1), -n+r^2+3r,r^2+r)$ is called a negative Latin square type partial difference set. In this paper, we generalize well-known negative Latin square type partial difference sets derived from the theory of cyclotomy. We use the partial difference sets in elementary abelian groups to generate analogous partial difference sets in nonelementary abelian groups of the form $(Z_p)^{4s} \times (Z_{p^s})^4$. It is believed that this is the first construction of negative Latin square type partial difference sets in nonelementary abelian $p$-groups where the $p$ can be any prime number. We also give a generalization of subsets of Type Q, partial difference sets consisting of one fourth of the nonidentity elements from the group, to nonelementary abelian groups. Finally, we give a similar product construction of negative Latin square type partial difference sets in the additive groups of $(F_q)^{4t+2}$ for an integer $t \geq 1$. This construction results in some new parameters of strongly regular graphs.