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.

There are exactly 35 inequivalent (36, 15, 6) difference sets in nine groups. Eight of the nine groups have a normal Sylow 3-subgroup. We give a straightforward spread construction which explains the 32 inequivalent difference sets in these eight groups. An interesting variation on this construction provides the three difference sets in the ninth group.

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.

A pair of planes, both projective or both affine, of the same order and on the same pointset are orthogoval if each line of one plane intersects each line of the other plane in at most two points. In this paper we prove new constructions for sets of mutually orthogoval planes, both projective and affine, and review known results that are equivalent to sets of more than two mutually orthogoval planes. We also discuss the connection between sets of mutually orthogoval planes and covering arrays.

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.

A $$(v,k,\lambda )$$ symmetric design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block. The designs associated to the symplectic difference sets introduced by Kantor (J Algebra 33:43–58, 1975) have the SDP. Parker (J Comb Theory Ser A 67:23–43, 1994) claimed that the symplectic design on 64 points is the only SDP design on 64 points admitting an abelian regular automorphism group (an abelian difference set). We show in this paper that there is an SDP design on 64 points that is not isomorphic to the symplectic design and yet admits the group $$C_8 \times C_4 \times C_2$$ as a regular automorphism group. This abelian difference set is the first in an infinite family of abelian difference sets whose designs have the SDP and yet are not isomorphic to the symplectic designs of the same order. We define a new method for establishing the non-isomorphism of the two families.

Let Gn denote the group C2n×C2n, where Ck is the cyclic group of order k. We give an algorithm for enumerating the regular nontrivial partial difference sets (PDS) in Gn. We use our algorithm to obtain all of these PDS in Gn for 2≤n≤9, and we obtain partial results for n=10 and n=11. Most of these PDS are new. For n≤4 we also identify group-inequivalent PDS. Our approach involves constructing tree diagrams and canonical colorings of these diagrams. Both the total number and the number of group-inequivalent PDS in Gn appear to grow super-exponentially in n. For n=9, a typical canonical coloring represents in excess of 10146 group-inequivalent PDS, and there are precisely 2520 reversible Hadamard difference sets.

We examine recent construction techniques of Hadamard difference sets in 2-groups, looking for an extension of orthogonal building sets to non-Abelian groups.

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 1978, Robert Kibler at the National Security Agency in Fort Meade, Maryland published a description of all noncyclic difference sets with $k < 20$. Kibler's decision to stop his extensive computer search for difference sets at block size 19 was motivated partly by the difficult barrier at $k=20$, the difference sets with parameters $(96,20,4)$. In this paper, we announce the completion of the search for all $(96,20,4)$ difference sets, relying on the computer software GAP and the work of numerous authors over the last few decades. The difference sets and the symmetric designs they create are summarized and links are provided to webpages which explicitly list the difference sets. In addition, we use these $(96,20,4)$ difference sets to construct all $(96, 20, 4, 4)$ and $(96, 19, 2, 4)$ partial difference sets and briefly look at the corresponding strongly regular graphs.

We give an algorithm for enumerating the regular nontrivial partial difference sets (PDS) in the group $G_n = C_{2^n}times C_{2^n}$. We use our algorithm to obtain all of these PDS in $G_n$ for $2leq nleq 9$, and we obtain partial results for $n=10$ and $n=11$. Most of these PDS are new. For $nle 4$ we also identify group-inequivalent PDS. Our approach involves constructing tree diagrams and canonical colorings of these diagrams. Both the total number and the number of group-inequivalent PDS in $G_n$ appear to grow super-exponentially in $n$. For $n=9$, a typical canonical coloring represents in excess of $10^{146}$ group-inequivalent PDS, and there are precisely $2^{520}$ reversible Hadamard difference sets.

We give an algorithm for enumerating the regular nontrivial partial difference sets (PDS) in the group G_n = C_2^n× C_2^n. We use our algorithm to obtain all of these PDS in G_n for 2≤ n≤ 9, and we obtain partial results for n=10 and n=11. Most of these PDS are new. For n≤ 4 we also identify group-inequivalent PDS. Our approach involves constructing tree diagrams and canonical colorings of these diagrams. Both the total number and the number of group-inequivalent PDS in G_n appear to grow super-exponentially in n. For n=9, a typical canonical coloring represents in excess of 10^146 group-inequivalent PDS, and there are precisely 2^520 reversible Hadamard difference sets.

We give an algorithm for enumerating the regular nontrivial partial difference sets (PDS) in the group $G_n = C_{2^n}\times C_{2^n}$. We use our algorithm to obtain all of these PDS in $G_n$ for $2\leq n\leq 9$, and we obtain partial results for $n=10$ and $n=11$. Most of these PDS are new. For $n\le 4$ we also identify group-inequivalent PDS. Our approach involves constructing tree diagrams and canonical colorings of these diagrams. Both the total number and the number of group-inequivalent PDS in $G_n$ appear to grow super-exponentially in $n$. For $n=9$, a typical canonical coloring represents in excess of $10^{146}$ group-inequivalent PDS, and there are precisely $2^{520}$ reversible Hadamard difference sets.

If a Hadamard difference set exists in H×K, where H is an abelian 2-group and K is a cyclic 3-group, then |H|>4|K|. Furthermore, Lander's conjecture holds for all Hadamard difference sets of order at most 529.

The sandpile group of a graph is a well-studied object that combines ideas from algebraic graph theory, group theory, dynamical systems, and statistical physics. A graph's sandpile group is part of a larger algebraic structure on the graph, known as its sandpile monoid. Most of the work on sandpiles so far has focused on the sandpile group rather than the sandpile monoid of a graph, and has also assumed the underlying graph to be undirected. A notable exception is the recent work of Babai and Toumpakari, which builds up the theory of sandpile monoids on directed graphs from scratch and provides many connections between the combinatorics of a graph and the algebraic aspects of its sandpile monoid. In this paper we primarily consider sandpile monoids on directed graphs, and we extend the existing theory in four main ways. First, we give a combinatorial classification of the maximal subgroups of a sandpile monoid on a directed graph in terms of the sandpile groups of certain easily-identifiable subgraphs. Second, we point out certain sandpile results for undirected graphs that are really results for sandpile monoids on directed graphs that contain exactly two idempotents. Third, we give a new algebraic constraint that sandpile monoids must satisfy and exhibit two infinite families of monoids that cannot be realized as sandpile monoids on any graph. Finally, we give an explicit combinatorial description of the sandpile group identity for every graph in a family of directed graphs which generalizes the family of (undirected) distance-regular graphs. This family includes many other graphs of interest, including iterated wheels, regular trees, and regular tournaments.

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.

We classify all circulant weighing matrices whose order and weight are products of powers of 2 and 3. In particular, we show that proper CW(v, 36)'s exist for all v equivalent to 0 (mod 48), all of which are new. (C) 2012 Published by Elsevier Inc.