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     

Strong difference families over arbitrary graphs

The concept of a strong difference family formally introduced in Buratti [J Combin Designs 7 (1999), 406–425] with the aim of getting group divisible designs with an automorphism group acting regularly on the points, is here extended for getting, more generally, sharply‐vertex‐transitive Γ‐decompositions of a complete multipartite graph for several kinds of graphs Γ. We show, for instance, that if Γ has e edges, then it is often possible to get a sharply‐vertex‐transitive Γ‐decomposition of K_{m × e} for any integer m whose prime factors are not smaller than the chromatic number of Γ. This is proved to be true whenever Γ admits an α‐labeling and, also, when Γ is an odd cycle or the Petersen graph or the prism T₅ or the wheel W₆. We also show that sometimes strong difference families lead to regular Γ‐decompositions of a complete graph. We construct, for instance, a regular cube‐decomposition of K_16m for any integer m whose prime factors are all congruent to 1 modulo 6. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 443–461, 2008     

Existence question for difference families and construction of some new families

We establish some properties of mixed difference families. We obtain some criteria for the existence of such families and a special kind of multipliers. Several methods are presented for the construction of difference families by using cyclotomy and genetic algorithms. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 256–270, 2004.     

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.     

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     

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     

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.     

Existence of (q,k, 1) difference families with q a prime power and k = 4, 5

The existence of a (q, k, 1) difference family in GF(q) has been completely solved for k = 3. For k = 4, 5 partial results have been given by Bose, Wilson, and Buratti. In this article, we continue the investigation and show that the necessary condition for the existence of a (q, k, 1) difference family in GF(q), i.e., q ≡ 1 (mod k(k − 1)) is also sufficient for k = 4, 5. For general k, Wilson's bound shows that a (q, k, 1) difference family in GF(q) exists whenever q ≡ 1 (mod k(k − 1)) and q > [k(k − 1)/2]^k(k−1). An improved bound on q is also presented. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 21–30, 1999     

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.     

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.     

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 cross-intersecting families

In this paper we investigate weighted cross-intersecting families: if α,β>0 are given constants, we want to find the maximum of α|A|+β|B| for A,B uniform cross-intersecting families. We determine the maximum sum, even if we have restrictions of the size of A.As corollaries, we will obtain some new bounds on the shadows and the shades of uniform families. We give direct proofs for these bounds, as well, and show that the theorems for cross-intersecting families also follow from these results.Finally, we will generalize the LYM inequality not only for cross-intersecting families, but also for arbitrary Sperner families.     

Solving isomorphism problems about 2‐designs from disjoint difference families

Recently, two new constructions of disjoint difference families in Galois rings were presented by Davis, Huczynska, and Mullen and Momihara. Both were motivated by a well‐known construction of difference families from cyclotomy in finite fields by Wilson. It is obvious that the difference families in the Galois ring and the difference families in the finite field are not equivalent. A related question, which is in general harder to answer, is whether the associated designs are isomorphic or not. In our case, this problem was raised by Davis, Huczynska, and Mullen. In this paper, we show that, in most cases, the 2‐ designs arising from the difference families in Galois rings and those arising from the difference families in finite fields are nonisomorphic by comparing their block intersection numbers.     

Constructions of partitioned difference families

Partitioned difference families (PDFs) arise from constructions of optimum constant composition codes. In this paper, a number of infinite classes of PDFs are constructed, based on known cyclotomic difference families in GF(q). A general approach which obtains PDFs from difference packings and coverings in abelian group is also presented.     

Recursive constructions for difference matrices and relative difference families

We present a new recursive construction for difference matrices whose application allows us to improve some results by D. Jungnickel. For instance, we prove that for any Abelian p-group G of type (n₁, n₂, …, n_t) there exists a (G, p^e, 1) difference matrix with e = Also, we prove that for any group G there exists a (G, p, 1) difference matrix where p is the smallest prime dividing |G|. Difference matrices are then used for constructing, recursively, relative difference families. We revisit some constructions by M. J. Colbourn, C. J. Colbourn, D. Jungnickel, K. T. Phelps, and R. M. Wilson. Combining them we get, in particular, the existence of a multiplier (G, k, λ)-DF for any Abelian group G of nonsquare-free order, whenever there exists a (p, k, λ)-DF for each prime p dividing |G|. Then we focus our attention on a recent construction by M. Jimbo. We improve this construction and prove, as a corollary, the existence of a (G, k, λ)-DF for any group G under the same conditions as above. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 165–182, 1998     

The reduction square root and perturbation for a class of strongly continuous operator families

We introduce the concept of a times C-second resolvent families and present the rela- tionship between a times C-resolvent families and a times C-second resolvent families. Moreover, the perturbation and square root for a times C-resolvent families are considered in this paper which generalize the counterparts of C-Cosine operator functions.     

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.     

Generic constructions for partitioned difference families with applications: a unified combinatorial approach

Partitioned difference families are an interesting class of discrete structures which can be used to derive optimal constant composition codes. There have been intensive researches on the construction of partitioned difference families. In this paper, we consider the combinatorial approach. We introduce a new combinatorial configuration named partitioned relative difference family, which proves to be very powerful in the construction of partitioned difference families. In particular, we present two general recursive constructions, which not only include some existing constructions as special cases, but also generate many new series of partitioned difference families. As an application, we use these partitioned difference families to construct several new classes of optimal constant composition codes.