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Signed difference sets have interesting applications in communications and coding theory. A (v , k , λ) -difference set in a finite group G of order v is a subset D of G with k distinct elements such that the expressions x y - 1 for all distinct two elements x , y ∈ D , represent each non-identity element in G exactly λ times. A (v , k , λ) -signed difference set is a generalization of a (v , k , λ) -difference set D, which satisfies all properties of D, but has a sign for each element in D. We will show some new existence results for signed difference sets by using partial difference sets, product methods, and cyclotomic classes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let p be a prime and let r , s be positive integers. In this paper, we prove that the Goormaghtigh equation $(x^m-1)/(x-1)=(y^n-1)/(y-1)$ , $x,y,m,n \in {\mathbb {N}}$ , $\min \{x,y\}>1$ , $\min \{m,n\}>2$ with $(x,y)=(p^r,p^s+1)$ has only one solution $(x,y,m,n)=(2,5,5,3)$. This result is related to the existence of some partial difference sets in combinatorics. [ABSTRACT FROM AUTHOR]

The concept of a modular difference set was originally motivated by the cognate notion of modular Hadamard matrices, which have been researched extensively. We initiate the study of the repetition-parameter set in a modular difference set, and we relate the repetition-parameter set to integer partitions and Diophantine equations. By example, we show how a computational study of integer partitions can improve the upper bound on the size of such repetition-parameter set. All previously known examples of modular difference sets in a direct product of groups are concerned with a product of just two groups. We present new constructions of modular difference sets in a direct product of n groups. These new constructions suggest that the size of the repetition-parameter set is intimately related to the group's structure. A generalization of difference sets, partial difference sets, and relative difference sets, modular difference sets have been used to construct both modular symmetric designs and equiangular tight frames in finite fields. [ABSTRACT FROM AUTHOR]

Bent functions f : V n → F p play an important role in constructing partial difference sets, where V n denotes an n-dimensional vector space over F p , p is an odd prime. In [2-3], the so-called vectorial dual-bent functions are considered to construct partial difference sets. In [2], Çeşmelioğlu et al. showed that for certain vectorial dual-bent functions F : V n → V s , the preimage set of 0 for F forms a partial difference set. In [3], Çeşmelioğlu et al. showed that for a class of Maiorana-McFarland vectorial dual-bent functions F : V n → F p s , the preimage set of the squares (non-squares) in F p s ∗ for F forms a partial difference set. In this paper, we further study vectorial dual-bent functions and partial difference sets. We prove that for certain vectorial dual-bent functions F : V n → F p s , the preimage set of the squares (non-squares) in F p s ∗ for F and the preimage set of any coset of some subgroup of F p s ∗ for F form partial difference sets. Furthermore, explicit constructions of partial difference sets are yielded from some (non-)quadratic vectorial dual-bent functions. In this paper, we illustrate that many results of using weakly regular p-ary bent functions to construct partial difference sets are special cases of our results. In [2], the authors considered weakly regular p-ary bent functions f with f (0) = 0 . They showed that if such a function f is an l-form with g c d (l - 1 , p - 1) = 1 for some integer 1 ≤ l ≤ p - 1 , then f is vectorial dual-bent. We prove that the converse also holds, which answers one open problem proposed in [3]. [ABSTRACT FROM AUTHOR]

The objective of this article is to broaden the understanding of the connections between bent functions and partial difference sets. Recently, the first two authors showed that the elements which a vectorial dual-bent function with certain additional properties maps to 0, form a partial difference set, which generalizes the connection between Boolean bent functions and Hadamard difference sets, and some later established connections between p-ary bent functions and partial difference sets to vectorial bent functions. We discuss the effects of coordinate transformations. As all currently known vectorial dual-bent functions F : F p n → F p s are linear equivalent to l-forms, i.e., to functions satisfying F (β x) = β l F (x) for all β ∈ F p s , we investigate properties of partial difference sets obtained from l-forms. We show that they are unions of cosets of F p s ∗ , which also can be seen as certain cyclotomic classes. We draw connections to known results on partial difference sets from cyclotomy. Motivated by experimental results, for a class of vectorial dual-bent functions from F p n to F p s , we show that the preimage set of the squares of F p s forms a partial difference set. This extends earlier results on p-ary bent functions. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of linear codes defined by characteristic functions of certain subsets of a finite field. We derive a sufficient and necessary condition for such a code to be a minimal linear code by a character-theoretical approach. We obtain new three-weight or four-weight minimal linear codes that do not satisfy the Ashikhmin-Barg condition by using partial difference sets. We show that our construction yields minimal linear codes that do not arise from cutting vectorial blocking sets, and also discuss their applications in secret sharing schemes. [ABSTRACT FROM AUTHOR]

A partial difference set S in a finite group G satisfying 1∉S and S=S−1 corresponds to an undirected strongly regular Cayley graph Cay(G,S). While the case when G is abelian has been thoroughly studied, there are comparatively few results when G is nonabelian. In this paper, we provide restrictions on the parameters of a partial difference set that apply to both abelian and nonabelian groups and are especially effective in groups with a nontrivial center. In particular, these results apply to p‐groups, and we are able to rule out the existence of partial difference sets in many instances. [ABSTRACT FROM AUTHOR]

In this paper, we consider regular automorphism groups of graphs in the RT2 family and the Davis‐Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non‐abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and Davis‐Xiang family and their amorphic abelian Cayley schemes to produce amorphic non‐abelian Cayley schemes. [ABSTRACT FROM AUTHOR]

Partial difference sets with parameters (v,k,λ,μ)=(v,(v−1)/2,(v−5)/4,(v−1)/4) are called Paley type partial difference sets. In this note, we prove that if there exists a Paley type partial difference set in an abelian group of order v, where v is not a prime power, then v=n4 or 9n4, n>1 an odd integer. In 2010, Polhill constructed Paley type partial difference sets in abelian groups with those orders. Thus, combining with the constructions of Polhill and the classical Paley construction using nonzero squares of a finite field, we completely answer the following question: "For which odd positive integers v>1, can we find a Paley type partial difference set in an abelian group of order v?" [ABSTRACT FROM AUTHOR]

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 2 t , r (3 t + 1) , - n + r 2 + 3 r , r 2 + r) , where r = 3 t - 1 + 1 (new for t ≥ 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. [ABSTRACT FROM AUTHOR]

In this paper we prove non-existence of nontrivial partial difference sets in Abelian groups of order 8p3, where p≥3 is a prime number. These groups seemed to have the potential of admitting at least two infinite families of PDSs, and even the smallest case, p=3, had been open for twenty years until settled recently by the authors and E. Neubert. Here, using the integrality and divisibility conditions for PDSs, we first describe all hypothetical parameter sets of nontrivial partial difference sets in these groups. Then we prove the non-existence of a PDS for each of these hypothetical parameter sets by combining a recent local multiplier result with some geometry and elementary number theory. [ABSTRACT FROM AUTHOR]

Strong external difference families (SEDFs) were introduced by Paterson and Stinson as a more restrictive version of external difference families. SEDFs can be used to produce optimal strong algebraic manipulation detection codes. We characterize the parameters (v,m,k,λ) of a nontrivial SEDF that is near-complete (satisfying v=km+1). We construct the first known nontrivial example of a (v,m,k,λ) SEDF having m>2. The parameters of this example are (243, 11, 22, 20), giving a near-complete SEDF, and its group is Z35. We provide a comprehensive framework for the study of SEDFs using character theory and algebraic number theory, showing that the cases m=2 and m>2 are fundamentally different. We prove a range of nonexistence results, greatly narrowing the scope of possible parameters of SEDFs. [ABSTRACT FROM AUTHOR]

For each non-constant q in the set of n-variable Boolean functions, the q-transform of a Boolean function f is related to the Hamming distances from f to the functions obtainable from q by nonsingular linear change of basis. Klapper conjectured that no Boolean function exists with its q-transform coefficients equal to ±2n/2 (such function is called q-bent) when q is non-affine balanced. In our early work, we only gave partial results to confirm this conjecture for small n. Here we prove thoroughly that the conjecture is true for all n by investigating the nonexistence of the partial difference sets in abelian groups with special parameters. We also introduce a new family of functions called (δ,q)-bent functions, which give a measurement of q-bentness. [ABSTRACT FROM AUTHOR]

Strong external difference families (SEDFs) and their generalizations GSEDFs and BGSEDFs in a finite abelian group G are combinatorial designs introduced by Paterson and Stinson (Discret Math 339: 2891-2906, 2016) and have applications in communication theory to construct optimal strong algebraic manipulation detection codes. In this paper we firstly present some general constructions of these combinatorial designs by using difference sets and partial difference sets in G. Then, as applications of the general constructions, we construct series of SEDF, GSEDF and BGSEDF in finite fields by using cyclotomic classes. Particularly, we present an (n,m,k,λ)=(243,11,22,20)-SEDF in (Fq,+)(q=35=243) by using the cyclotomic classes of order 11 in Fq which answers an open problem raised in Paterson and Stinson (2016). [ABSTRACT FROM AUTHOR]

In this note we prove the non-existence of two types of partial difference sets in Abelian groups of order 216 . This completes the classification of parameters for which a partial difference set of size at most 100 exists in an Abelian group. [ABSTRACT FROM AUTHOR]

In this article we provide a complete classification of regular partial difference sets in Abelian groups of order $$4p^2$$ , p an odd prime. It turns out that the known examples are the only examples. These are, up to complements, the trivial examples, the PCP examples, and a sporadic example in $$\mathbb {Z}_2^2\times \mathbb {Z}_3^2$$ . [ABSTRACT FROM AUTHOR]

We introduce the concept of a disjoint partial difference family (DPDF) and an external partial difference family (EPDF), a natural generalization of the much-studied disjoint difference family (DDF), external difference family (EDF) and partial difference set (PDS). We establish properties and indicate connections to other recently-studied combinatorial structures. We show how DPDFs and EPDFs may be formed from PDSs, and present various cyclotomic constructions for DPDFs and EPDFs. As part of this, we develop a unified cyclotomic framework, which yields some known results on PDSs, DDFs and EDFs as special cases. [ABSTRACT FROM AUTHOR]

In this paper we present a new cyclotomic family of partial difference sets, which includes a family presented in Fernández-Alcober et al. (2010). The argument rests on a general procedure for constructing cyclotomic difference sets or partial difference sets in Galois domains due to Ott (2015). [ABSTRACT FROM AUTHOR]

In this article we generalize a theorem of Benson (J Algebra 15:443-454, 1970) for generalized quadrangles to strongly regular graphs, deriving numerical restrictions on the number of fixed vertices and the number of vertices mapped to adjacent vertices under an automorphism. We then use this result to develop a few new techniques to study regular partial difference sets (PDS) in Abelian groups. Ma (Des Codes Cryptogr 4:221-261, 1994) provided a list of parameter sets of regular PDS with $$k\le 100$$ in Abelian groups for which existence was known or had not been excluded. In particular there were 32 parameter sets for which existence was not known. Ma (J Stat Plan Inference 62:47-56, 1997) excluded 13 of these parameter sets. As an application of our results we here exclude the existence of a regular partial difference set for all but two of the undecided parameter sets from Ma's list. [ABSTRACT FROM AUTHOR]

A partial difference set S in a finite group G such that S = S-1 and 1 ∉ S corresponds to an undirected strongly regular Cayley graph Cay(G,S). Very few examples of PDSs are known, and there are especially few known in nonabelian groups. In this paper, a partial difference set S with S = S-1 and 1 ∉ S is constructed for each extraspecial group of order p3 and exponent p2, where p is an odd prime. [ABSTRACT FROM AUTHOR]

The only known skew-translation generalised quadrangles (STGQ) having order ( q , q ) , with q even, are translation generalised quadrangles. Equivalently, the only known groups G of order q 3 , q even, admitting an Ahrens–Szekeres (AS-)configuration are elementary abelian. In this paper we prove results in the theory of STGQ giving (i) new structural information for a group G admitting an AS-configuration, (ii) a classification of the STGQ of order ( 8 , 8 ) , and (iii) a classification of the STGQ of order ( q , q ) for odd q (using work of Ghinelli and Yoshiara). [ABSTRACT FROM AUTHOR]

There exist few examples of negative Latin square type partial difference sets (NLST PDSs) in nonabelian groups. We present a list of 176 inequivalent NLST PDSs in 48 nonisomorphic, nonabelian groups of order 64. These NLST PDSs form 8 nonisomorphic strongly regular graphs. These PDSs were constructed using a combination of theoretical techniques and computer search, both of which are described. The search was run exhaustively on 212/267 nonisomorphic groups of order 64. [ABSTRACT FROM AUTHOR]

Almost difference sets are an interesting subject of combinatorics, and have applications in many areas of engineering such as CDMA communications, error correcting codes and cryptography. The objective of this paper is to present some new constructions of almost difference sets, together with several results on the equivalence relation. [ABSTRACT FROM AUTHOR]

Using tools from algebraic graph theory, we prove that for every odd prime $$p$$ there is just one proper loop of order $$2p$$ such that the automorphism group of the corresponding rank 5 association scheme contains a regular subgroup of order $$4p^2$$. Although our results were initially obtained on the basis of theoretical generalization of a tremendous number of computer algebra experiments, our final computer-free proof uses quite elementary arguments from group theory and combinatorics. In this paper we provide this computer-free proof, as well as discuss further suggested use of our methods. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields by using union of cyclotomic classes of of order , where is a prime and m a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers. [Copyright &y& Elsevier]

Abstract: A partial difference set with parameters is said to be of Paley type. In this paper, we give a recursive theorem that for all odd constructs Paley partial difference sets in certain groups of order and . We are also able to construct Paley–Hadamard difference sets of the Stanton–Sprott family in groups of order when is a prime power and when is a prime power. Many of these are new parameters for such difference sets, and also give new Hadamard designs and matrices. [ABSTRACT FROM AUTHOR]

Abstract: We present results related to vectorial plateaued functions and mappings whose derivatives are -to-1 functions. The results in this note generalize facts about almost perfect nonlinear and almost bent functions. We investigate the connection between plateaued and -to-1 functions. We show that functions which are both plateaued and differentially uniform give rise to partial difference sets. [Copyright &y& Elsevier]

Abstract: If R is a homomorphic image of a finite Frobenius local ring, there is a known construction that produces Latin square type partial difference sets (PDS) in . By a simple construction, we show that every finite ring is a homomorphic image of a finite Frobenius ring and every finite local ring is a homomorphic image of a finite Frobenius local ring. Consequently, Latin square type PDS can be constructed in for any finite local ring R, where the additive group can be any finite abelian p-group. [Copyright &y& Elsevier]

Abstract: Applying results from partial difference sets, quadratic forms, and recent results of Brouwer and Van Dam, we construct the first known amorphic association scheme with negative Latin square-type graphs and whose underlying set is a nonelementary abelian 2-group. We give a simple proof of a result of Hamilton that generalizes Brouwer''s result. We use multiple distinct quadratic forms to construct amorphic association schemes with a large number of classes. [Copyright &y& Elsevier]

We present three constructions of partial difference sets (PDS) using different types of finite local rings. The first construction uses homomorphic images of finite Frobenius local rings and generalizes a previous result by the author. The second construction uses finite Frobenius local rings. The third construction uses finite (noncommutative) chain rings and generalizes a recent construction of partial difference sets by Leung and Ma. The three constructions provide many new PDS in nonelementary abelian groups. [Copyright &y& Elsevier]

Bent functions andpartial difference sets have been constructed from finite principalideal local rings. In this paper, the constructions are generalizedto finite quasi-Frobenius local rings. Let R bea finite quasi-Frobenius local ring with maximal ideal M.Bent functions and certain partial difference sets on M } M are extended to R } R. [ABSTRACT FROM AUTHOR]

Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an important role in the theory. On the other hand, Polhill (2010) gave a construction of Paley type partial difference sets (conference graphs) based on a special system of building blocks, called a covering extended building set, and proved that there exists a Paley type partial difference set in an abelian group of order 9 i v 4 for any odd positive integer v > 1 and any i = 0 , 1. His result covers all orders of abelian non p -groups in which Paley type partial difference sets exist. In this paper, we give new constructions of strongly regular Cayley graphs on abelian groups by extending the theory of building blocks. The constructions are large generalizations of Polhill's construction. In particular, we show that for a positive integer m and elementary abelian groups G i , i = 1 , 2 , ... , s , of order q i 4 such that 2 m | q i + 1 , there exists a decomposition of the complete graph on the abelian group G = G 1 × G 2 × ⋯ × G s by strongly regular Cayley graphs with negative Latin square type parameters (u 2 , c (u + 1) , − u + c 2 + 3 c , c 2 + c) , where u = q 1 2 q 2 2 ⋯ q s 2 and c = (u − 1) / m. Such strongly regular decompositions were previously known only when m = 2 or G is a p -group. Moreover, we find one more new infinite family of decompositions of the complete graphs by Latin square type strongly regular Cayley graphs. Thus, we obtain many strongly regular graphs with new parameters. [ABSTRACT FROM AUTHOR]

A partial geometry S admitting an abelian Singer group G is called of rigid type if all lines of S have a trivial stabilizer in G. In this paper, we show that if a partial geometry of rigid type has fewer than 1000000 points it must be the Van Lint–Schrijver geometry or be a hypothetical geometry with 1024 or 4096 or 194481 points, which provides evidence that partial geometries of rigid type are very rare. Along the way we also exclude an infinite set of parameters that originally seemed very promising for the construction of partial geometries of rigid type (as it contains the Van Lint–Schrijver parameters as its smallest case and one of the other cases we cannot exclude as the second member of this parameter family). We end the paper with a conjecture on this type of geometries. [ABSTRACT FROM AUTHOR]

Partial difference sets (for short, PDSs) with parameters (n 2 , r (n − ϵ) , ϵ n + r 2 − 3 ϵ r , r 2 − ϵ r) are called Latin square type (respectively negative Latin square type) PDSs if ϵ = 1 (respectively ϵ = − 1). In this paper, we will give restrictions on the parameter r of a (negative) Latin square type partial difference set in an abelian group of non-prime power order a 2 b 2 , where gcd ⁡ (a , b) = 1 , a > 1 , and b is an odd positive integer ≥3. Very few general restrictions on r were previously known. Our restrictions are particularly useful when a is much larger than b. As an application, we show that if there exists an abelian negative Latin square type PDS with parameter set (9 p 4 s , r (3 p 2 s + 1) , − 3 p 2 s + r 2 + 3 r , r 2 + r) , 1 ≤ r ≤ 3 p 2 s − 1 2 , p ≡ 1 (mod 4) a prime number and s is an odd positive integer, then there are at most three possible values for r. For two of these three r values, J. Polhill gave constructions in 2009 [10]. [ABSTRACT FROM AUTHOR]

Let G n denote the group C 2 n × C 2 n , where C k is the cyclic group of order k. We give an algorithm for enumerating the regular nontrivial partial difference sets (PDS) in G 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 1 0 146 group-inequivalent PDS, and there are precisely 2 520 reversible Hadamard difference sets. [ABSTRACT FROM AUTHOR]