Existence Of 4-SCGDDs Of Type G~n And Their Applications

A group divisible design of type gn, denoted by k-GDD, is a triple (X, g, B), where X is a set of gn elements (called points), G is a partition of X into n subsets (called groups) of size g, B is a family of k-subsets of X (called blocks) such that each pair of points from different groups occurs in exactly one block and any pair of points from the same group does not ouccur in any block. Letπbe a permutation over X. If iπ(B)={π(p)|p∈B}∈B for each B∈B, we callπan automorphism of the k-GDD. Furthermore, ifπpermute elements in each group G∈g in a\G\cycle, the k-GDD is called semi-cyclic, denoted by k-SCGDD.Semi-cyclic group divisible designs have been found to be closely related to optical or-thogonal codes (OOCs), an important kind of optical address codes in optical code division multiple access (OCDMA) system. A k-SCGDD of type gn can be uesed to construct an optimal (n x g, k,1) AM-OPPW 2-D OOC. Moreover, a k-SCGDD of type gn is equivalent to a generalized Bhaskar Rao design GBRD(n, k, g; Zg).For the existence problem of a k-SCGDD of type gn, the case where k = 3 has been solved completely by Robert. P. Gallant, Zhike Jiang and Alan C. H. Ling; the case where k = 2, the case where k = 4, n = 4, g is even and the case where k = 4, g is odd and n≥5, n∈N have been solved completely by J. Wang and J. Yin; the case where k = 4, n = 4, g is odd is solved partly by J. Wang and J. Yin. In this paper, the case where k = 4, n≥5 and g is even is dealt with, and the following main result is proved by means of some direct constructions and recursive constructions. Let g be even, n>≥5, n∈N, Eo={(