On the Non-existence of Semiregular Relative Difference Sets in Groups with All Sylow Subgroups Cyclic

We show that a group with all Sylow subgroups cyclic (other than ) cannot contain a normal semiregular relative difference set (RDSs). We also give a new proof that dihedral groups cannot contain (normal) semiregular RDSs either.     

A New Family of Relative Difference Sets in 2-Groups

We recursively construct a new family of ( 2^6d+4, 8, 2^6d+4, 2^6d+1) semi-regular relative difference sets in abelian groups G relative to an elementary abelian subgroup U. The initial case d = 0 of the recursion comprises examples of (16, 8, 16, 2) relative difference sets for four distinct pairs (G, U).     

Constructions of Semi-regular Relative Difference Sets

J. A. Davis, J. Jedwab, and M. Mowbray (1998, Des. Codes Cryptogr.13, 131–146) gave two new constructions for semi-regular relative difference sets (RDSs). They asked if the two constructions could be unified. In this paper, we show that the two constructions are closely related. In fact, the second construction should be viewed as an extension of the first. Furthermore, we generalize the second construction to obtain new RDSs.     

Linking systems of difference sets

A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All previous constructive results for linking systems of difference sets are restricted to 2‐groups. We use an elementary projection argument to show that neither the McFarland/Dillon nor the Spence construction of difference sets can give rise to a linking system of difference sets in non‐2‐groups. We make a connection to Kerdock and bent sets, which provides large linking systems of difference sets in elementary abelian 2‐groups. We give a new construction for linking systems of difference sets in 2‐groups, taking advantage of a previously unrecognized connection with group difference matrices. This construction simplifies and extends prior results, producing larger linking systems than before in certain 2‐groups, new linking systems in other 2‐groups for which no system was previously known, and the first known examples in nonabelian groups.     

A group extensions approach to relative difference sets

A new approach to (normal) relative difference sets (RDSs) is presented and applied to give a new method for recursively constructing infinite families of semiregular RDSs. Our main result (Theorem 7.1) shows that any metabelian semiregular RDS gives rise to an infinite family of metabelian semiregular RDSs. The new method is applied to identify several new infinite families of non‐abelian semiregular RDSs, and new methods for constructing generalized Hadamard matrices are given. The techniques employed are derived from the general theory of group extensions. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 279–298, 2004.     

Semi-regular relative difference sets with large forbidden subgroups

Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters (m,n,m,m/n) in groups of non-prime-power orders. Let p be an odd prime. We prove that there does not exist a (2p,p,2p,2) relative difference set in any group of order 2p², and an abelian (4p,p,4p,4) relative difference set can only exist in the group . On the other hand, we construct a family of non-abelian relative difference sets with parameters (4q,q,4q,4), where q is an odd prime power greater than 9 and . When q=p is a prime, p>9, and , the (4p,p,4p,4) non-abelian relative difference sets constructed here are genuinely non-abelian in the sense that there does not exist an abelian relative difference set with the same parameters.     

Relative difference sets,graphs and inequivalence of functions between groups

For cryptographic purposes, we want to find functions with both low differential uniformity and dissimilarity to all linear functions and to know when such functions are essentially different. For vectorial Boolean functions, extended affine equivalence and the coarser Carlet–Charpin–Zinoviev (CCZ) equivalence are both used to distinguish between nonlinear functions. It remains hard to tell when CCZ equivalent functions are EA‐inequivalent. This paper presents a framework for solving this problem in full generality, for functions between arbitrary finite groups. This common framework is based on relative difference sets (RDSs). The CCZ and EA equivalence classes of perfect nonlinear (PN) functions are each derived, by quite different processes, from equivalence classes of splitting semiregular RDSs. By generalizing these processes, we obtain a much strengthened formula for all the graph equivalences which define the EA equivalence class of a given function, amongst those which define its CCZ equivalence class. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 260–273, 2010     

A Family of Semi-Regular Divisible Difference Sets

We give a construction of semi-regular divisible difference sets with parametersm = p2a(r–1)+2b (pr – 1)/(p – 1), n = pr, k = p(2a+1)(r–1)+2b (pr – 1)/(p – 1)1 = p(2a+1)(r–1)+2b (pr–1 – 1)/(p-1), 2 = p2(a+1)(r–1)–r+2b (pr – 1)/(p – 1)where p is a prime and r a + 1.     

Strongly Regular Semi-Cayley Graphs

We consider strongly regular graphs = (V, E) on an even number, say 2n, of vertices which admit an automorphism group G of order n which has two orbits on V. Such graphs will be called strongly regular semi-Cayley graphs. For instance, the Petersen graph, the Hoffman–Singleton graph, and the triangular graphs T(q) with q 5 mod 8 provide examples which cannot be obtained as Cayley graphs. We give a representation of strongly regular semi-Cayley graphs in terms of suitable triples of elements in the group ring Z G. By applying characters of G, this approach allows us to obtain interesting nonexistence results if G is Abelian, in particular, if G is cyclic. For instance, if G is cyclic and n is odd, then all examples must have parameters of the form 2n = 4s ² + 4s + 2, k = 2s ² + s, = s ² – 1, and = s ²; examples are known only for s = 1, 2, and 4 (together with a noncyclic example for s = 3). We also apply our results to obtain new conditions for the existence of strongly regular Cayley graphs on an even number of vertices when the underlying group H has an Abelian normal subgroup of index 2. In particular, we show the nonexistence of nontrivial strongly regular Cayley graphs over dihedral and generalized quaternion groups, as well as over two series of non-Abelian 2-groups. Up to now these have been the only general nonexistence results for strongly regular Cayley graphs over non-Abelian groups; only the first of these cases was previously known.     

Divisible designs and semi-regular relative difference sets from additive Hadamard cocycles

Additive Hadamard cocycles are a natural generalization of presemifields. In this paper, we study divisible designs and semi-regular relative difference sets obtained from additive Hadamard cocycles. We show that the designs obtained from additive Hadamard cocycles are flag transitive. We introduce a new product construction of Hadamard cocycles. We also study additive Hadamard cocycles whose divisible designs admit a polarity in which all points are absolute. Our main results include generalizations of a theorem of Albert and a theorem of Hiramine from presemifields to additive Hadamard cocycles. At the end, we generalize Maiorana-McFarland?s construction of bent functions to additive Hadamard cocycles.     

Negative Latin square type partial difference sets in nonabelian groups of order 64

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.     

Abelian Difference Sets Without Self-conjugacy

We obtain some results that are useful to the study of abelian difference sets and relative difference sets in cases where the self-conjugacy assumption does not hold. As applications we investigate McFarland difference sets, which have parameters of the form v=q^d+1( q^d+ q^d-1 +...+ q+2) ,k=q^d( q^d+q^d-1+...+q+1) , = q^d ( q^(d-1)+q^(d-2)+...+q+1), where q is a prime power andd a positive integer. Using our results, we characterize those abelian groups that admit a McFarland difference set of order k- = 81. We show that the Sylow 3-subgroup of the underlying abelian group must be elementary abelian. Our results fill two missing entries in Kopilovich's table with answer no.     

Difference Sets Corresponding to a Class of Symmetric Designs

We study difference sets with parameters(v, k, ) = (p ^s(r ^2m - 1)/(r - 1), p ^s-1 r ^2m-2 r - 1)r ^{2m -2}, where r = r ^s - 1)/(p - 1) and p is a prime. Examples for such difference sets are known from a construction of McFarland which works for m = 1 and all p,s. We will prove a structural theorem on difference sets with the above parameters; it will include the result, that under the self-conjugacy assumption McFarland's construction yields all difference sets in the underlying groups. We also show that no abelian .160; 54; 18/-difference set exists. Finally, we give a new nonexistence prove of (189, 48, 12)-difference sets in Z ₃ × Z ₉ × Z ₇.     

A Difference Set in \left( {\mathbb{Z}/4\mathbb{Z}} \right)^3 \times \mathbb{Z}/5\mathbb{Z}

We present a (320, 88, 24)-difference set in , the existence of which was previously open. This new difference set improves a theorem of Davis-Jedwab with the removal of the exceptional case. It also enables us to state a theorem of Schmidt on Davis-Jedwab difference sets more neatly.     

Coprime ordering of cyclic planar difference sets

Motivated by a question about uniform dessins d’enfants, it is conjectured that every cyclic planar difference set of prime power order m≠4 can be cyclically ordered such that the difference of every pair of neighbouring elements is coprime to the module v?m²+m+1. We prove that this is the case whenever the number ω(v) of different prime divisors of v is less than or equal to 3. To achieve this we consider a graph related to the difference set and show that it is Hamiltonian.     

Relative ( pn , p , pn , n )-difference sets with GCD ( p , n )=1

Let p be an odd prime. We first get some non-existence and structural results on (pn,p,pn,n) relative difference sets with gcd(p,n)=1 through a group ring approach. We then give a construction of (p(p+1),p,p(p+1),p+1) relative difference sets with p a Mersenne prime. Supported by National Natural Science Foundation of China (10331030).     

Normal cirulant graphs with noncyclic regular subroups

We prove that any circulant graph of order n with connection set S such that n and the order of ?(S), the subgroup of ? that fixes S set‐wise, are relatively prime, is also a Cayley graph on some noncyclic group, and shows that the converse does not hold in general. In the special case of normal circulants whose order is not divisible by 4, we classify all such graphs that are also Cayley graphs of a noncyclic group, and show that the noncyclic group must be metacyclic, generated by two cyclic groups whose orders are relatively prime. We construct an infinite family of normal circulants whose order is divisible by 4 that are also normal Cayley graphs on dihedral and noncyclic abelian groups. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Paley type partial difference sets in non <Emphasis Type="Italic">p</Emphasis>-groups

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 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 when is a prime power. These are new parameters for such difference sets.      

A bound for the orders of the torsion groups of surfaces with c 1 2 = 2 - 1

We shall give a bound for the orders of the torsion groups of minimal algebraic surfaces of general type whose first Chern numbers are twice the Euler characteristics of the structure sheaves minus 1, where the torsion group of a surface is the torsion part of the Picard group. Namely, we shall show that the order is at most 3 if the Euler characteristic is 2, that the order is at most 2 if the Euler characteristic is greater than or equal to 3, and that the order is 1 if the Euler characteristic is greater than or equal to 7.