A Construction of Difference Sets

In this paper, we will give a construction of a family of -difference sets in thegroup , where q is any power of 2, K is any group with and G is an abelian 2-group of order which contains anelementary abelian subgroup of index 2.     

On Sylow Subgroups of Abelian Affine Difference Sets

An n-subsetD of a group G of order is called an affine difference set of G relativeto a normal subgroup N of G of order if the list of differences containseach element of G-N exactly once and no elementof N. It is a well-known conjecture that if Dis an affine difference set in an abelian group G,then for every prime p, the Sylow p-subgroupof G is cyclic. In Arasu and Pott [1], it was shownthat the above conjecture is true when . In thispaper we give some conditions under which the Sylow p-subgroupof G is cyclic.     

Difference Sets and Hyperovals

We construct three infinite families of cyclic difference sets, using monomial hyperovals in a desarguesian projective plane of even order. These difference sets give rise to cyclic Hadamard designs, which have the same parameters as the designs of points and hyperplanes of a projective geometry over the field with two elements. Moreover, they are substructures of the Hadamard design that one can associate with a hyperoval in a projective plane of even order.     

New Families of Semi-Regular Relative Difference Sets

We give two constructions for semi-regular relative difference sets (RDSs) in groups whose order is not a prime power, where the order u of the forbidden subgroup is greater than 2. No such RDSs were previously known. We use examples from the first construction to produce semi-regular RDSs in groups whose order can contain more than two distinct prime factors. For u greater than 2 these are the first such RDSs, and for u=2 we obtain new examples.     

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.     

Difference Sets with n = 2pm

Let D be a (v,k,) difference set over an abelian group G with even n = k - . Assume that t N satisfies the congruences t q _i ^fi (mod exp(G)) for each prime divisor q_i of n/2 and some integer f_i. In [4] it was shown that t is a multiplier of D provided that n > , (n/2, ) = 1 and (n/2, v) = 1. In this paper we show that the condition n > may be removed. As a corollary we obtain that in the case of n= 2p^a when p is a prime, p should be a multiplier of D. This answers an open question mentioned in [2].     

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.     

Cyclic Relative Difference Sets and their p-Ranks

By modifying the constructions in Helleseth et al. [10] and No [15], we construct a family of cyclic ((q ^3k –1)/(q–1), q–1, q ^3k–1, q ^3k–2) relative difference sets, where q=3 ^e . These relative difference sets are liftings of the difference sets constructed in Helleseth et al. [10] and No [15]. In order to demonstrate that these relative difference sets are in general new, we compute p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when q=3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.     

Constructions of Partial Difference Sets and Relative DifferenceSets Using Galois Rings

We use Galois rings to construct partial difference sets and relative difference sets in non-elementary abelianp-groups. As an example, we also use Galois ringG R(4, 2) to construct a (96,20,4) difference set in Z₄ × Z₄ × Z₆.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

Impossibility of a Certain Cyclotomic Equation with Applicationsto Difference Sets

Under certain conditions, we show the nonexistence ofan element in the p-th cyclotomicfield over , that satisfies . As applications, we establish the nonexistence ofsome difference sets and affine difference sets.     

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).     

Foldings of Difference Sets in Abelian Groups

 In this paper, we show that under some conditions the existence of a difference set in G implies the existence of another difference set with the same parameters in G′, where G and G′ are abelian groups of the same order. This explains why there are more difference sets in abelian groups of low exponent and high rank than in those of high exponent and low rank. Received: September 1, 1997 / Revised: March 24, 1998     

Multiplicative Difference Sets via Additive Characters

We use Fourier analysis on the additive group of to give an alternative proof of the recent theorem of Maschietti and to prove recent conjectures of No, Chung and Yun and No, Golomb, Gong, Lee and Gaal on difference sets in the multiplicative group of , m odd. Along the ay e prove a stronger form of a celebrated theorem of Welch on the 3-valued cross-correlation of maximal length sequences.     

Two Generalized Constructions of Relative Difference Sets

We give two generalizations of some known constructions of relative difference sets. The first one is a generalization of a construction of RDS by Chen, Ray-Chaudhuri and Xiang using the Galois ring GR(4, m). The second one generalizes a construction of RDS by Ma and Schmidt from the setting of chain rings to a setting of more general rings.     

One (96,20,4)-symmetric Design and related Nonabelian Difference Sets

New (96,20,4)-symmetric design has been constructed, unique under the assumption of an automorphism group of order 576 action. 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 five nonabelian groups of order 96. None of them belongs to the class of groups that allow the application of so far known construction (McFarland, Dillon) for McFarland difference sets.AMS lassification: 05B05     

Bent Functions, Partial Difference Sets, and Quasi-Frobenius Local Rings

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.     

Spherical Designs and Generalized Sum-Free Sets in Abelian Groups

We extend the concepts of sum-freesets and Sidon-sets of combinatorial number theory with the aimto provide explicit constructions for spherical designs. We calla subset S of the (additive) abelian group G t-free if for all non-negative integers kand l with k+l t, the sum of k(not necessarily distinct) elements of S does notequal the sum of l (not necessarily distinct) elementsof S unless k=l and the two sums containthe same terms. Here we shall give asymptotic bounds for thesize of a largest t-free set in Z _n,and for t 3 discuss how t-freesets in Z _n can be used to constructspherical t-designs.     

A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2^2d+2, 2^2d+1±2 ^d , 2^2d ±2 ^d ). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2 ^d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2^2d+2 with exponent 2 ^d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.     

On the p-Ranks and Characteristic Polynomials of Cyclic Difference Sets

In this paper, the p-ranks and characteristic polynomials of cyclic difference sets are derived by expanding the trace expressions of their characteristic sequences. Using this method, it is shown that the 3-ranks and characteristic polynomials of the Helleseth–Kumar–Martinsen (HKM) difference set and the Lin difference set can be easily obtained. Also, the p-rank of a Singer difference set is reviewed and the characteristic polynomial is calculated using our approach.     

New Constructions of Disjoint Distinct Difference Sets

New constructions of regular disjoint distinct difference sets (DDDS) are presented. In particular, multiplicative and additive DDDS are considered.