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.     

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     

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.     

Weighted external difference families and R-optimal AMD codes

In this paper, we provide a mathematical framework for characterizing AMD codes that are R-optimal. We introduce a new combinatorial object, the reciprocally-weighted external difference family (RWEDF), which corresponds precisely to an R-optimal weak AMD code. This definition subsumes known examples of existing optimal codes, and also encompasses combinatorial objects not covered by previous definitions in the literature. By developing structural group-theoretic characterizations, we exhibit infinite families of new RWEDFs, and new construction methods for known objects such as near-complete EDFs. Examples of RWEDFs in non-abelian groups are also discussed.     

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.     

Constructions of almost difference families

Almost difference families (ADFs) are a useful generalization of almost difference sets (ADSs). In this paper, we present some constructive techniques to obtain ADFs and establish a number of infinite classes of ADFs. Our results can be regarded as a generalization of the known difference families. It is clear that ADFs give partially balance incomplete block designs which arise in a natural way in many combinatorial and statistical problems.     

Partial Geometric Difference Families

We introduce the notion of a partial geometric difference family as a variation on the classical difference family and a generalization of partial geometric difference sets. We study the relationship between partial geometric difference families and both partial geometric designs and difference families, and show that partial geometric difference families give rise to partial geometric designs. We construct several infinite families of partial geometric difference families using Galois rings and the cyclotomy of Galois fields. From these partial geometric difference families, we generate a list of infinite families of partial geometric designs and directed strongly regular graphs.     

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.     

Recursive constructions of secure codes and hash families using difference function families

To protect copyrighted digital data against piracy, codes with different secure properties such as frameproof codes, secure frameproof codes, codes with identifiable parent property (IPP codes), traceability codes (TA codes) are introduced. In this paper, we study these codes together with related combinatorial objects called separating and perfect hash families. We introduce for the first time the notion of difference function families and use these difference function families to give generalized recursive techniques that can be used for any kind of secure codes and hash families. We show that some previous recursive techniques are special cases of these new techniques.     

Constructing difference families through an optimization approach: Six new BIBDs

In this paper we formulate the construction of difference families as a combinatorial optimization problem. A tabu search algorithm is used to find an optimal solution to the optimization problem for various instances of difference families. In particular, we construct six new difference families which lead to an equal number of new balanced incomplete block designs with parameters: (49, 98, 18, 9, 3), (61, 122, 20, 10, 3), (46, 92, 20, 10, 4), (45, 90, 22, 11, 5), (85, 255, 24, 8, 2) and (34, 85, 30, 12, 10). © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 261–273, 2000     

New partial geometric difference sets and partial geometric difference families

Using Galois rings and Galois fields, we construct several infinite classes of partial geometric difference sets, and partial geometric difference families, with new parameters. Furthermore, these partial geometric difference sets (and partial geometric difference families) correspond to new infinite families of directed strongly regular graphs. We also discuss some of the links between partially balanced designs, 2-adesigns (which were recently coined by Cunsheng Ding in “Codes from Difference Sets” (2015)), and partial geometric designs, and make an investigation into when a 2-adesign is a partial geometric design.     

Partial geometric difference sets and partial geometric difference families

The concept of a partial geometric difference set (or $1\frac{1}{2}$-difference set) was introduced by Olmez in 2014. Recently, Nowak, Olmez and Song introduced the notion of a partial geometric difference family, which generalizes both the classical difference family and the partial geometric difference set. It was shown that partial geometric difference sets and partial difference families give rise to partial geometric designs. In this paper, a number of new infinite families of partial geometric difference sets and partial geometric difference families are constructed. From these partial geometric difference sets and difference families, we generate a list of infinite families of partial geometric designs.     

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.     

广义差集与几乎完美序列

提出了广义差集的概念,并且给出了广义差集的一些初等性质.从应用的角度讲,广义差集就是使得其±1特征序列的自相关函数是(最多)三值的一种组合结构.因此,广义差集不仅仅是在概念(理论)上的推广,它还具有深层次的应用背景.事实上,给出了一些广义差集,它不是可分差集,也不是相对差集.同时也给出了一类广义差集存在的一些必要条件,使得这些广义差集对应的±1特征序列成为几乎完美序列.并举例说明本文中的方法是有效的.     

A generalization of combinatorial designs and related codes

In this paper we study a certain generalization of combinatorial designs related to almost difference sets, namely the t-adesign, which was coined by Ding (Codes from difference sets, 2015). It is clear that 2-adesigns are partially balanced incomplete block designs which naturally arise in many combinatorial and statistical problems. We discuss some of their basic properties and give several constructions of 2-adesigns (some of which correspond to new almost difference sets and some to new almost difference families), as well as two constructions of 3-adesigns. We discuss basic properties of the incidence matrices and make an initial investigation into the codes which they generate. We find that many of the codes have good parameters in the sense they are optimal or have relatively high minimum distance.     

Perfect difference families,perfect difference matrices,and related combinatorial structures

The existence problems of perfect difference families with block size k, k=4,5, and additive sequences of permutations of length n, n=3,4, are two outstanding open problems in combinatorial design theory for more than 30 years. In this article, we mainly investigate perfect difference families with block size k=4 and additive sequences of permutations of length n=3. The necessary condition for the existence of a perfect difference family with block size 4 and order v, or briefly (v, 4,1)‐PDF, is v≡1(mod12), and that of an additive sequence of permutations of length 3 and order m, or briefly ASP (3, m), is m≡1(mod2). So far, (12t+1,4,1)‐PDFs with t<50 are known only for t=1,4−36,41,46 with two definiteexceptions of t=2,3, and ASP (3, m)'s with odd 3<m<200 are known only for m=5,7,13−29,35,45,49,65,75,85,91,95,105,115,119,121,125,133,135,145,147,161,169,175,189,195 with two definite exceptions of m=9,11. In this article, we show that a (12t+1,4,1)‐PDF exists for any t⩽1,000 except for t=2,3, and an ASP (3, m) exists for any odd 3<m<200 except for m=9,11 and possibly for m=59. The main idea of this article is to use perfect difference families and additive sequences of permutations with “holes”. We first introduce the concepts of an incomplete perfect difference matrix with a regular hole and a perfect difference packing with a regular difference leave, respectively. We show that an additive sequence of permutations is in fact equivalent to a perfect difference matrix, then describe an important recursive construction for perfect difference matrices via perfect difference packings with a regular difference leave. Plenty of perfect difference packings with a desirable difference leave are constructed directly. We also provide a general recursive construction for perfect difference packings, and as its applications, we obtain extensive recursive constructions for perfect difference families, some via incomplete perfect difference matrices with a regular hole. Examples of perfect difference packings directly constructed are used as ingredients in these recursive constructions to produce vast numbers of perfect difference families with block size 4. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 415–449, 2010     

Strongly regular graphs associated with ternary bent functions

We prove a new characterization of weakly regular ternary bent functions via partial difference sets. Partial difference sets are combinatorial objects corresponding to strongly regular graphs. Using known families of bent functions, we obtain in this way new families of strongly regular graphs, some of which were previously unknown. One of the families includes an example in [N. Hamada, T. Helleseth, A characterization of some {3v₂+v₃,3v₁+v₂,3,3}-minihypers and some [15,4,9;3]-codes with B₂=0, J. Statist. Plann. Inference 56 (1996) 129-146], which was considered to be sporadic; using our results, this strongly regular graph is now a member of an infinite family. Moreover, this paper contains a new proof that the Coulter-Matthews and ternary quadratic bent functions are weakly regular.     

A Construction for Resolvable Designs and Its Generalizations

 In [14], D.K. Ray-Chaudhuri and R.M. Wilson developed a construction for resolvable designs, making use of free difference families in finite fields, to prove the asymptotic existence of resolvable designs with index unity. In this paper, generalizations of this construction are discussed. First, these generalizations, some of which require free difference families over rings in which there are some units such that their differences are still units, are used to construct frames, resolvable designs and resolvable (modified) group divisible designs with index not less than one. Secondly, this construction method is applied to resolvable perfect Mendelsohn designs. Thirdly, cardinalities of such sets of units are investigated. Finally, composition theorems for free difference families via difference matrices are described. They can be utilized to produce some new examples of resolvable designs.     

Existence and non-existence results for strong external difference families

We consider strong external difference families (SEDFs); these are external difference families satisfying additional conditions on the patterns of external differences that occur, and were first defined in the context of classifying optimal strong algebraic manipulation detection codes. We establish new necessary conditions for the existence of $(n,m,k,\lambda )$-SEDFs; in particular giving a near-complete treatment of the $\lambda =2$ case. For the case $m=2$, we obtain a structural characterization for partition type SEDFs (of maximum possible $k$ and $\lambda$), showing that these correspond to Paley partial difference sets. We also prove a version of our main result for generalized SEDFs, establishing non-trivial necessary conditions for their existence.     

On the measure of intersecting families,uniqueness and stability

Let t≥1 be an integer and let A be a family of subsets of {1,2,…,n} every two of which intersect in at least t elements. Identifying the sets with their characteristic vectors in {0,1} ⁿ we study the maximal measure of such a family under a non uniform product measure. We prove, for a certain range of parameters, that the t-intersecting families of maximal measure are the families of all sets containing t fixed elements, and that the extremal examples are not only unique, but also stable: any t-intersecting family that is close to attaining the maximal measure must in fact be close in structure to a genuine maximum family. This is stated precisely in Theorem 1.6. We deduce some similar results for the more classical case of Erdős-Ko-Rado type theorems where all the sets in the family are restricted to be of a fixed size. See Corollary 1.7. The main technique that we apply is spectral analysis of intersection matrices that encode the relevant combinatorial information concerning intersecting families. An interesting twist is that part of the linear algebra involved is done over certain polynomial rings and not in the traditional setting over the reals. A crucial tool that we use is a recent result of Kindler and Safra [22] concerning Boolean functions whose Fourier transforms are concentrated on small sets. Research supported in part by the Israel Science Foundation, grant no. 0329745.