New Constructions of Divisible Designs

A construction is given for a (p2a(p+1),p2,p2a+1(p+1),p2a+1,p2a(p+1)) (p a prime) divisible difference set in the group H×Z2pa+1 where H is any abelian group of order p+1. This can be used to generate a symmetric semi-regular divisible design; this is a new set of parameters for λ1≠0, and those are fairly rare. We also give a construction for a (pa−1+pa−2+…+p+2,pa+2, pa(pa+pa−1+…+p+1), pa(pa−1+…+p+1), pa−1(pa+…+p2+2)) divisible difference set in the group H×Zp2×Zap. This is another new set of parameters, and it corresponds to a symmetric regular divisible design. For p=2, these parameters have λ1=λ2, and this corresponds to the parameters for the ordinary Menon difference sets

Matrix constructions of divisible designs

AbstractWe present two new constructions of group divisible designs. We use skew-symmetric Hadamard matrices and certain strongly regular graphs together with (v, k, λ)-designs. We include many examples, in particular several new series of divisible difference sets

Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three

The concept of group divisible codes, a generalization of group divisible designs with constant block size, is introduced in this paper. This new class of codes is shown to be useful in recursive constructions for constant-weight and constant-composition codes. Large classes of group divisible codes are constructed which enabled the determination of the sizes of optimal constant-composition codes of weight three (and specified distance), leaving only four cases undetermined. Previously, the sizes of constant-composition codes of weight three were known only for those of sufficiently large length.Comment: 13 pages, 1 figure, 4 table

HFR Code: A Flexible Replication Scheme for Cloud Storage Systems

Fractional repetition (FR) codes are a family of repair-efficient storage codes that provide exact and uncoded node repair at the minimum bandwidth regenerating point. The advantageous repair properties are achieved by a tailor-made two-layer encoding scheme which concatenates an outer maximum-distance-separable (MDS) code and an inner repetition code. In this paper, we generalize the application of FR codes and propose heterogeneous fractional repetition (HFR) code, which is adaptable to the scenario where the repetition degrees of coded packets are different. We provide explicit code constructions by utilizing group divisible designs, which allow the design of HFR codes over a large range of parameters. The constructed codes achieve the system storage capacity under random access repair and have multiple repair alternatives for node failures. Further, we take advantage of the systematic feature of MDS codes and present a novel design framework of HFR codes, in which storage nodes can be wisely partitioned into clusters such that data reconstruction time can be reduced when contacting nodes in the same cluster.Comment: Accepted for publication in IET Communications, Jul. 201