On group divisible covering designs

(3, Λ)-group Divisible Covering Designs

Let U, g, k and A be positive integers with u :::: k. A (k, A)-grOUp divisible covering design ((k, A)-GDCD) with type gU is a A-cover of pairs by k-tuples of a gu-set X with u holes of size g, which are disjoint and spanning. The covering number, C(k, A; gil), is the minimum number of blocks in a (k, A)-GDCD of type gUo In this paper, the detennination ofllie fimction C(3, A; gil) begun by [6]...

Construction of Group Divisible Designs

Optimal constant weight covering codes and nonuniform group divisible 3-designs with block size four

Let Kq(n, w, t, d) be the minimum size of a code over Zq of length n, constant weight w, such that every word with weight t is within Hamming distance d of at least one codeword. In this article, we determine Kq(n, 4, 3, 1) for all n ≥ 4, q = 3, 4 or q = 2m + 1 with m ≥ 2, leaving the only case (q, n) = (3, 5) in doubt. Our construction method is mainly based on the auxiliary designs, H-frames,...

Divisible designs with dual translation group

Many different divisible designs are already known. Some of them possess remarkable automorphism groups, so called dual translation groups. The existence of such an automorphism group enables us to characterize its associated divisible design as being isomorphic to a substructure of a finite affine space. AMS Classification: 05B05, 05B30, 20B25, 51N10

Resolvable Group Divisible Designs with Large Groups

We prove that the necessary divisibility conditions are sufficient for the existence of resolvable group divisible designs with a fixed number of sufficiently large groups. Our method combines an application of the Rees product construction with a streamlined recursion based on incomplete transversal designs. With similar techniques, we also obtain new results on decompositions of complete mult...