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.     

The Existence of (v,4,1) Disjoint Difference Families with a Prime Power

A(υ,κ,λ,)difference family((υ,κ,λ,)-DF in short)over an abelian group G of order v,is a collection F = {B_i|i∈I}ofκ-subsets of G,called base blocks,such that any nonzero element of G can be represented in preciselyλways as a difference of two elements lying in some base blocks in F.A(υ,κ,λ,)-DDF is a difference family with disjoint blocks.In this paper,by using Weil's theorem on character sum estimates,it is proved that there exists a(p~n,4,1)-DDF,where p≡1(rood 12)is a prime number and n≥1.     

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.     

Difference systems of sets based on cosets partitions

Difference systems of sets （DSS） are combinatorial configurations that arise in connection with code synchronization. This paper proposes a new method to construct DSSs, which uses known DSSs to partition some of the cosets of Zv relative to subgroup of order k, where v = km is a composite number. As applications, we obtain some new optimal DSSs.     

相对差集和组合集的构造

利用Galois环、Bent函数、Gaolis环上的部分指数和等技巧,构造了指数不超过4的有限交换群上的分裂型相对差集和一类非分裂型组合集.     

Constructions for Anonymous Secret Sharing Schemes Using Combinatorial Designs

In an anonymous secret sharing scheme the secret can be reconstructed without knowledge ofwhich participants hold which shares.In this paper some constructions of anonymous secret sharing schemeswith 2 thresholds by using combinatorial designs are given.Let v(t,w,q)denote the minimum size of the setof shares of a perfect anonymous(t,w)threshold secret sharing scheme with q secrets.In this paper we provethat v(t,w,q)=(q)if t and w are fixed and that the lower bound of the size of the set of shares in[4]is notoptimal under certain condition.     

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     

External Difference Families from Finite Fields

External difference families (EDFs) are a type of combinatorial designs that originated from cryptography. Many combinatorial objects are closely related to EDFs, such as difference sets, difference families, almost difference sets, and difference systems of sets. Constructing EDFs is thus of significance in theory and practice. In this paper, earlier ideas of constructing EDFs proposed by Chang and Ding (2006), and Huang and Wu (2009), are further explored. Consequently, new infinite classes of EDFs are obtained and some previously known results are extended.     

Internal and external partial difference families and cyclotomy

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.     

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.     

Strong difference families,difference covers,and their applications for relative difference families

In (M. Buratti, J Combin Des 7:406–425, 1999), Buratti pointed out the lack of systematic treatments of constructions for relative difference families. The concept of strong difference families was introduced to cover such a problem. However, unfortunately, only a few papers consciously using the useful concept have appeared in the literature in the past 10 years. In this paper, strong difference families, difference covers and their connections with relative difference families and optical orthogonal codes are discussed.      

Direct Constructions for General Families of Cyclic Mutually Nearly Orthogonal Latin Squares

Two Latin squares and , of even order n with entries , are said to be nearly orthogonal if the superimposition of L on M yields an array in which each ordered pair , and , occurs at least once and the ordered pair occurs exactly twice. In this paper, we present direct constructions for the existence of general families of three cyclic mutually orthogonal Latin squares of orders , , and . The techniques employed are based on the principle of Methods of Differences and so we also establish infinite classes of “quasi‐difference” sets for these orders.     

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.     

Disjoint congruence classes and a timetabling application

We consider a combinatorial problem motivated by a special simplified timetabling problem for subway networks. Mathematically the problem is to find (pairwise) disjoint congruence classes modulo certain given integers; each such class corresponds to the arrival times of a subway line of a given frequency. For a large class of instances we characterize when such disjoint congruence classes exist and how they may be determined. We also study a generalization involving a minimum distance requirement between congruence classes, and a comparison of different frequency families in terms of their “efficiency”. Finally, a general method based on integer programming is also discussed.     

Computational Results for Regular Difference Systems of Sets Attaining or Being Close to the Levenshtein Bound

Difference systems of sets (DSSs) are combinatorial structures arising in connection with code synchronization that were introduced by Levenshtein in 1971, and are a generalization of cyclic difference sets. In this paper, we consider a collection of m‐subsets in a finite field of prime order to be a regular DSS for an integer m, and give a lower bound on the parameter ρ of the DSS using cyclotomic numbers. We show that when we choose ‐subsets from the multiplicative group of order e, the lower bound on ρ is independent of the choice of subsets. In addition, we present some computational results for DSSs with block sizes and , whose parameter ρ attains or comes close to the Levenshtein bound for .     

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     

3-Wise Exactly 1-Intersecting Families of Sets

Let f(l, t, n) be the maximal size of a family such that any l2 sets of have an exactly t1-element intersection. If l3, it trivially comes from [8] that the optimal families are trivially intersecting (there is a t-element core contained by all the members of the family). Hence it is easy to determine Let g(l,t,n) be the maximal size of an l-wise exaclty t-intersecting family that is not trivially t-intersecting. We give upper and lower bounds which only meet in the following case: g(3, 1, n) = n^2/3(1 + o(1)).     

Families intersecting on an interval

We shall be interested in the following Erd?s-Ko-Rado-type question. Fix some set B⊂[n]={1,2,…,n}. How large a subfamily A of the power set P[n] can we find such that the intersection of any two sets in A contains a cyclic translate (modulo n) of B? Chung, Graham, Frankl and Shearer have proved that, in the case where B=[t] is a block of length t, we can do no better than taking A to consist of all supersets of B. We give an alternative proof of this result, which is in a certain sense more ‘direct’.     

Disjoint Unions of Topological Spaces and Choice

We find properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice.