A nonexistence result for abelian menon difference sets using perfect binary arrays

A Menon difference set has the parameters (4N ² ,2N ² -N, N ² -N). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian group contains a Menon difference set, wherep is an odd prime, |K|=p , andp ^j–1 (mod exp (H)) for somej. Using the viewpoint of perfect binary arrays we prove thatK must be cyclic. A corollary is that there exists a Menon difference set in the abelian group , where exp(H)=2 or 4 and |K|=3, if and only ifK is cyclic.This work is partially supported by NSA grant # MDA 904-92-H-3057 and by NSF grant # NCR-9200265. The author thanks the Mathematics Department, Royal Holloway College, University of London for its hospitality during the time of this researchThis work is partially supported by NSA grant # MDA 904-92-H-3067     

Reversible relative difference sets

We investigate nontrivial (m, n, k, )-relative difference sets fixed by the inverse. Examples and necessary conditions on the existence of relative difference sets of this type are studied.     

On the existence of difference sets in groups of order 96

The correspondence between a (96,20,4) symmetric design having regular automorphism group and a difference set with the same parameters has been used to obtain difference sets in groups of order 96. Starting from eight such symmetric designs constructed by the tactical decomposition method, 55 inequivalent (96,20,4) difference sets are distinguished. Thereby the existence of difference sets in 22 nonabelian groups of order 96 is proved.     

A note on difference sets in dihedral groups

It is proved that there are no nontrivial difference sets in dihedral groups of order 4p^t, where p is a prime, t > 0 is a positive integer. Received: 22 February 2002     

New necessary conditions on Paley‐type partial difference sets in abelian groups

Let be an abelian group of order , where are distinct odd prime numbers. In this paper, we prove that if contains a regular Paley‐type partial difference set (PDS), then for any is congruent to 3 modulo 4 whenever is odd. These new necessary conditions further limit the specific order of an abelian group in which there can exist a Paley‐type PDS. Our result is similar to a result on abelian Hadamard (Menon) difference sets proved by Ray‐Chaudhuri and Xiang in 1997.     

On central difference sets in certain non-abelian 2-groups

In this note, we define the class of finite groups of Suzuki type, which are non-abelian groups of exponent 4 and class 2 with special properties. A group G of Suzuki type with |G|=2^2s always possesses a non-trivial difference set. We show that if s is odd, G possesses a central difference set, whereas if s is even, G has no non-trivial central difference set.     

On a relation between a cyclic relative difference set associated with the quadratic extensions of a finite field and the Szekeres difference sets

Letq 3 (mod 4) be a prime power and put . We consider a cyclic relative difference set with parametersq ²–1,q, 1,q–1 associated with the quadratic extension GF(q²)/GF((q). The even part and the odd part of the cyclic relative difference set taken modulon are supplementary difference sets. Moreover it turns out that their complementary subsets are identical with the Szekeres difference sets. This result clarifies the true nature of the Szekeres difference sets. We prove these results by using the theory of the relative Gauss sums.     

Difference sets and inverting the difference operator

For a setA of non-negative numbers, letD(A) (the difference set ofA) be the set of nonnegative differences of elements ofA, and letD ^k be thek-fold iteration ofD. We show that for everyk, almost every set of non-negative integers containing 0 arises asD ^k (A) for someA. We also give sufficient conditions for a setA to be the unique setX such that 0X andD ^k (X)=D ^k (A). We show that for eachm there is a setA such thatD(X)=D(A) has exactly 2 ^m solutionsX with 0X.This work was supported by grants DMS 92-02833 and DMS 91-23478 from the National Science Foundation. The first author acknowledges the support of the Hungarian National Science Foundation under grants, OTKA 4269, and OTKA 016389, and the National Security Agency (grant No. MDA904-95-H-1045).Lee A. Rubel died March 25, 1995. He is very much missed by his coauthors.     

Difference sets without squares

A sequence of natural numbersA=a ₁,a ₂, ... is constructed such that noa _i −a _j is a square and there are >x ^0,73 a _i 's belowx.     

A problem of Erdős on abelian groups

The following theorem is proved. Ifa ₁,a ₂, ...a _n are nonzero elements inZ _n , and are not all equal, then ε₁ a ₁+ε₂ a ₂+...+ε _n a _n =0 has at leastn solutions with ε _i =0 or 1.     

Countably infinite quasi-convex sets in some locally compact abelian groups

We produce a class of countably infinite quasi-convex sets (sequences converging to zero) in the circle group T and in the group J₂ of 2-adic integers determined by sequences of integers satisfying a mild lacunarity condition. We also extend our results to the group R of real numbers. All these quasi-convex sets have a stronger property: Every infinite (necessarily) symmetric subset containing 0 is still quasi-convex.     

Relative difference sets fixed by inversion and Cayley graphs

Using graph theoretical technique, we present a construction of a (30,2,29,14)-relative difference set fixed by inversion in the smallest finite simple group—the alternating group A₅. To our knowledge this is the first example known of relative difference sets in the finite simple groups with a non-trivial forbidden subgroup. A connection is then established between some relative difference sets fixed by inversion and certain antipodal distance-regular Cayley graphs. With the connection, several families of antipodal distance-regular Cayley graphs which are coverings of complete graphs are presented.     

Path-closed sets

Given a digraphG = (V, E), call a node setT ⊆V path-closed ifv, v′ εT andw εV is on a path fromv tov′ impliesw εT. IfG is the comparability graph of a posetP, the path-closed sets ofG are the convex sets ofP. We characterize the convex hull of (the incidence vectors of) all path-closed sets ofG and its antiblocking polyhedron inR ^v , using lattice polyhedra, and give a minmax theorem on partitioning a given subset ofV into path-closed sets. We then derive good algorithms for the linear programs associated to the convex hull, solving the problem of finding a path-closed set of maximum weight sum, and prove another min-max result closely resembling Dilworth’s theorem.     

Minimal sets in finite rings

We describe how to calculate the (, )-minimal sets in any finite ring.     

On IP* sets and central sets

IP^* sets and central sets are subsets of which are known to have rich combinatorial structure. We establish here that structure is significantly richer that was previously known. We also establish that multiplicatively central sets have rich additive structure. The relationship among IP^* sets, central sets, and corresponding dynamical notions are also investigated.The authors gratefully acknowledge support from the National Science Foundation (USA) via grants DMS-9103056 and DMS-9025025 respectively.     

The existence of Cartan connections and geometrizable principle bundles

The aim of this article is to give necessary and sufficient conditions for the existence of Cartan connections on principal bundles. For this purpose we recall the definition of abstract soldering forms and introduce the notion of geometrizable principle bundles.Received: 13 November 2003