Disjoint difference families and their applications

Difference sets and their generalisations to difference families arise from the study of designs and many other applications. Here we give a brief survey of some of these applications, noting in particular the diverse definitions of difference families and the variations in priorities in constructions. We propose a definition of disjoint difference families that encompasses these variations and allows a comparison of the similarities and disparities. We then focus on two constructions of disjoint difference families arising from frequency hopping sequences and showed that they are in fact the same. We conclude with a discussion of the notion of equivalence for frequency hopping sequences and for disjoint difference families

Near-complete external difference families

We introduce and explore near-complete external difference families, a partitioning of the nonidentity elements of a group so that each nonidentity element is expressible as a difference of elements from distinct subsets a fixed number of times. We show that the existence of such an object implies the existence of a near-resolvable design. We provide examples and general constructions of these objects, some of which lead to new parameter families of near-resolvable designs on a non-prime-power number of points. Our constructions employ cyclotomy, partial difference sets, and Galois rings.PostprintPeer reviewe

ON THE EMBEDDING OF GROUPS AND DESIGNS IN A DIFFERENCE BLOCK DESIGN

A difference BIBD is a balanced incomplete block design on a group which isconstructed by transferring a regular perfect difference system by a subgroup of its point set. There is an obvious bijection between these BIBDs and some copies of their point sets as two sets. In this paper, we investigate the algebraic structure of these block designs by definning a group-isomorphism between them and their point sets. It has done by defning some relations between the independent-graphs of difference BIBDs and some Cayley graphs of their point sets. It is shown that some Cayley graphs are embedded in the independent-graph of difference BIBDs as a spanning sub-graphs. Due to find these relations, we find out a configuration ordering on these BIBDs, also we achieve some results about the classification of these BIBDs. All in this paper are on difference BIBDs with even numbers of the points

Six Constructions of Difference Families

In this paper, six constructions of difference families are presented. These constructions make use of difference sets, almost difference sets and disjoint difference families, and give new point of views of relationships among these combinatorial objects. Most of the constructions work for all finite groups. Though these constructions look simple, they produce many difference families with new parameters. In addition to the six new constructions, new results about intersection numbers are also derived

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, �)-SEDFs; in particular giving a near-complete treatment of the � = 2 case. For the case m = 2, we obtain a structural characterization for partition type SEDFs (of maximum possible k and �), 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

Existence and non-existence results for strong external difference families

The first author is supported by a Research Incentive Grant from The Carnegie Trust for the Universities of Scotland (Grant No. 70582).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 , λ) -SEDFs; in particular giving a near-complete treatment of the λ = 2 case. For the case m = 2 , we obtain a structural characterization for partition type SEDFs (of maximum possible k and λ), 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.PostprintPeer reviewe