Indivisible plexes in latin squares

In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v ₁ and v ₂ with |v ₁ − v ₂| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁵ or 3⁷, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁹. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ≤ 3¹¹.      

Two constructions of (v, (v − 1)/2, (v − 3)/2) difference families

In this article, two constructions of (v, (v ? 1)/2, (v ? 3)/2) difference families are presented. The first construction produces both cyclic and noncyclic difference families, while the second one gives only cyclic difference families. The parameters of the second construction are new. The difference families presented in this article can be used to construct Hadamard matrices. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 164–171, 2008     

A New Method for Constructing Williamson Matrices

For every prime power q 1 (mod 4) we prove the existence of (q; x, y)-partitions of GF(q) with q=x²+4y² for some x, y, which are very useful for constructing SDS, DS and Hadamard matrices. We discuss the transformations of (q; x,y)-partitions and, by using the partitions, construct generalized cyclotomic classes which have properties similar to those of classical cyclotomic classes. Thus we provide a new construction for Williamson matrices of order q².The research supported by NSF of China (No. 10071029).     

Constructions of Nested Partial Difference Sets with Galois Rings

There have been several recent constructions of partial difference sets (PDSs) using the Galois rings for p a prime and t any positive integer. This paper presents constructions of partial difference sets in where p is any prime, and r and t are any positive integers. For the case where 2$$ " align="middle" border="0"> many of the partial difference sets are constructed in groups with parameters distinct from other known constructions, and the PDSs are nested. Another construction of Paley partial difference sets is given for the case when p is odd. The constructions make use of character theory and of the structure of the Galois ring , and in particular, the ring × . The paper concludes with some open related problems.     

Exponential number of inequivalent difference sets in ℤ

Kantor [ 5 ] proved an exponential lower bound on the number of pairwise inequivalent difference sets in the elementary abelian group of order 2^2s+2. Dillon [ 3 ] generalized a technique of McFarland [ 6 ] to provide a framework for determining the number of inequivalent difference sets in 2‐groups with a large elementary abelian direct factor. In this paper, we consider the opposite end of the spectrum, the rank 2 group ? , and compute an exponential lower bound on the number of pairwise inequivalent difference sets in this group. In the process, we demonstrate that Dillon difference sets in groups ? can be constructed via the recursive construction from [ 2 ] and we show that there are exponentially many pairwise inequivalent difference sets that are inequivalent to any Dillon difference set. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 249–259, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10046     

Hadamard and Conference Matrices

We discuss new constructions of Hadamard and conference matrices using relative difference sets. We present the first example of a relative -difference set where n – 1 is not a prime power.     

Planar cyclic difference packings

A planar cyclic difference packing modulo v is a collection D = {d₁, d₂,…,d_k} of distinct residues modulo v such that any residue α ≢ 0 (mod v) has at most one representation as a difference d_i − d_j (mod v). This paper develops various constructions of such designs and for a fixed positive integer v presents upper and lower bounds on Ψ (v), the maximal number of elements that a planar cyclic difference packing modulo v can contain. This paper also presents the results of calculating Ψ (v) for v ≤ 144, including the fact that 134 is the smallest value of v for which the elementary upper bound of exceeds Ψ (v) by two. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 426–434, 2000     

Random intersections of thick Cantor sets

Let , be Cantor sets embedded in the real line, and let , be their respective thicknesses. If , then it is well known that the difference set is a disjoint union of closed intervals. B. Williams showed that for some , it may be that is as small as a single point. However, the author previously showed that generically, the other extreme is true; contains a Cantor set for all in a generic subset of . This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if , then contains a Cantor set for almost all in .

     

Constructions of External Difference Families and Disjoint Difference Families

External difference families (EDFs) are a type of new combinatorial designs originated from cryptography. In this paper, some earlier ideas of recursive and cyclotomic constructions of combinatorial designs are extended, and a number of classes of EDFs and disjoint difference families are presented. A link between a subclass of EDFs and a special type of (almost) difference sets is set up.     

Unified Treatment of Algebraic and Geometric Difference by a New Difference Scheme and its Continuity Properties

We introduce a new operation for the difference of two sets A and C of R ⁿ depending on a parameter . This new operation may yield as special cases the classical difference and the Minkowski difference, if the sets A and C are closed, convex sets, if int(C) is nonempty, and if A or C bounded. Continuity properties with respect to both the operands and the parameter of this operation are studied. Lipschitz properties of the Minkowski difference between two sets of a normed vector space are proved in the bounded case as well as in the unbounded case without condition on the dimension of the space.     

Special subsets of difference sets with particular emphasis on skew Hadamard difference sets

This article introduces a new approach to studying difference sets via their additive properties. We introduce the concept of special subsets, which are interesting combinatorial objects in their own right, but also provide a mechanism for measuring additive regularity. Skew Hadamard difference sets are given special attention, and the structure of their special subsets leads to several results on multipliers, including a categorisation of the full multiplier group of an abelian skew Hadamard difference set. We also count the number of ways to write elements as a product of any number of elements of a skew Hadamard difference set.      

Codebooks from almost difference sets

In direct spread CDMA systems, codebooks meeting the Welch bounds are used to distinguish among the signals of different users. Recently, constructions of codebooks with difference sets meeting Welch’s bound on the maximum cross-correlation amplitude were developed. In this paper, a generic construction of codebooks using almost difference sets is considered and several classes of codebooks nearly meeting the Welch bound are obtained. The parameters of the codebooks constructed in this paper are new.      

Cyclotomic constructions of external difference families and disjoint difference families

External difference families (EDFs) are a type of new combinatorial designs originated from cryptography. Some results had been obtained by Chang and Ding, the connection between EDFs and disjoint difference families (DDFs) was also established. In this paper, further cyclotomic constructions of EDFs and DDFs are presented, and several classes of EDFs and DDFs are obtained. Answers to problems 1 and 4 by Chang and Ding are also given. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 333–341, 2009     

New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

In this paper, for a prime power q, new cyclic difference sets with Singer para- meters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed by using q-ary sequences (d-homogeneous functions) of period q ⁿ –1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets with Singer parameters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed from the ternary sequences of period q ⁿ –1 with ideal autocorrelation introduced by Helleseth, Kumar, and Martinsen.     

Toward the classification of biangular harmonic frames

Equiangular tight frames (ETFs) and biangular tight frames (BTFs) – sets of unit vectors with basis-like properties whose pairwise absolute inner products admit exactly one or two values, respectively – are useful for many applications. A well-understood class of ETFs are those which manifest as harmonic frames – vector sets defined in terms of the characters of finite abelian groups – because they are characterized by combinatorial objects called difference sets.This work is dedicated to the study of the underlying combinatorial structures of harmonic BTFs. We show that if a harmonic frame is generated by a divisible difference set, a partial difference set or by a special structure with certain Gauss summing properties – all three of which are generalizations of difference sets that fall under the umbrella term “bidifference set” – then it is either a BTF or an ETF. However, we also show that the relationship between harmonic BTFs and bidifference sets is not as straightforward as the correspondence between harmonic ETFs and difference sets, as there are examples of bidifference sets that do not generate harmonic BTFs. In addition, we study another class of combinatorial structures, the nested divisible difference sets, which yields an example of a harmonic BTF that is not generated by a bidifference set.     

A note on products of Relative Difference Sets

Relative Difference Sets with the parameters k = n have been constructed many ways (see (Davis, forthcoming; Elliot and Butson 1966; and Jungnickel 1982)). This paper proves a result on building new RDS by taking products of others (much like (Dillon 1985)), and this is applied to several new examples (primarily involving (p ⁱ, p ^j, p ⁱ, p ^i–j)).     

Some remarks on orders of projective planes,planar difference sets and multipliers

In this paper we investigate how finite group theory, number theory, together with the geometry of substructures can be used in the study of finite projective planes. Some remarks concerning the function v(x)= x ² + x + 1are presented, for example, how the geometry of a subplane affects the factorization of v(x). The rest of this paper studies abelian planar difference sets by multipliers.Partially supported by NSA grant MDA904-90-H-1013.     

Constructions of optimal difference systems of sets

Difference systems of sets (DSSs) are combinatorial configurations which were introduced in 1971 by Levenstein for the construction of codes for synchronization. In this paper, we present two kinds of constructions of difference systems of sets by using disjoint difference families and a special type of difference sets, respectively. As a consequence, new infinite classes of optimal DSSs are obtained.