A structural classification of SR group divisible designs

A simple method of construction of a semi-regular (SR) group divisible design from another SR group divisible design is given. Using this method, 111 available SR designs from Clatworthy (1973) and John and Turner (1977) are systematically classified into 20 classes. This procedure may produce new nonisomorphic solutions for known designs.     

Spreads and group divisible designs

By removing the components of at-spread of a finite projective spacePG(d, q) from each hyperplane ofPG(d, q), the blocks of a regular group divisible design are obtained We characterize geometrict-spreads as thoset-spreads which are such that the dual of is also a group divisible design.     

Composite construction of group divisible designs

Two new methods of constructing group divisible designs are given. In particular, a new resolvable solution for the SR 39 is presented.     

Linked systems of symmetric group divisible designs

We introduce the concept of linked systems of symmetric group divisible designs. The connection with association schemes is established, and as a consequence we obtain an upper bound on the number of symmetric group divisible designs which are linked. Several examples of linked systems of symmetric group divisible designs are provided.     

Elliptic semiplanes and group divisible designs with orthogonal resolutions

In this paper we prove that there exists an elliptic semiplaneS(v, k, m) withk –m 2 if and only if there exists a group divisible design GDD _k ((k –m)(k – 1);k –m; 0, 1) withm pairwise orthogonal resolutions. As an example of this theorem, we construct an elliptic semiplaneW(45, 7, 3) and show thatW is isomorphic to the elliptic semiplaneS(45, 7, 3) given by R. D. Baker.     

Asymptotic existence theorems for frames and group divisible designs

In this paper, we establish an asymptotic existence theorem for group divisible designs of type mn with block sizes in any given set K of integers greater than 1. As consequences, we will prove an asymptotic existence theorem for frames and derive a partial asymptotic existence theorem for resolvable group divisible designs.     

An analogue of group divisible designs for Moore graphs

We study graphs whose adjacency matrix S of order n satisfies the equation S + S² = J ? K + kI, where J is a matrix of order n of all 1's, K is the direct sum on $\text{n}\text{l}$ matrices of order l of all 1's, and I is the identity matrix. Moore graphs are the only solutions to the equation in the case l = 1 for which K = I. In the case k = l we can obtain Moore graphs from a solution S by a bordering process analogous to obtaining (ν, κ, λ)-designs from some group divisible designs. Other parameters are rare. We are able to find one new interesting graph with parameters k = 6, l = 4 on n = 40 vertices. We show that it has a transitive automorphism group isomorphic to C₄ × S₅.     

Embeddings of resolvable group divisible designs with block size 3

It is proved in this paper that an RGD(3, g;v) can be embedded in an RGD(3, g;u) if and only if , , , v ≥ 3g, u ≥ 3v, and (g,v) ≠ (2,6),(2,12),(6,18).     

Near-symmetric divisible designs

We construct some classes of -near-symmetric divisible designs by permutation group methods. We also define and study Paley divisible designs which generalize the well-known class of Paley 2-designs.Dedicated to professor Giuseppe Tallini     

Resolvable group divisible designs with block size four and index three

In this paper, we investigate the existence of resolvable group divisible designs (RGDDs) with block size four, group-type hⁿ and index three. The necessary conditions for the existence of such a design are n?4 and hn≡0. These necessary conditions are shown to be sufficient except for (h,n)∈{(2,4),(2,6)} and possibly excepting (h,n)=(2,54).     

Modified group divisible designs with block size 5 and even index

In this paper it is shown that for λ even, (5,λ)-MGDDs of type mⁿ exists whenever the known necessary conditions are satisfied, with a finite number of exceptions.     

Resolvable group divisible designs with block size four and general index

In this paper, we investigate the existence of resolvable group divisible designs (RGDDs) with block size four, group-type hⁿ and general index λ. The necessary conditions for the existence of such a design are n≥4, and . These necessary conditions are shown to be sufficient for all λ≥2, with the definite exceptions of (λ,h,n)∈{(3,2,6)}∪{(2j+1,2,4):j≥1}. The known existence result for λ=1 is also improved.     

Incomplete group divisible designs with block size four and general index

In this paper, we investigate the existence of incomplete group divisible designs (IGDDs) with block size four, group-type (g, h) u and general index λ. The necessary conditions for the existence of such a design are that u ≥ 4, g ≥ 3h, λg(u 1) ≡ 0 (mod 3), λ(g h)(u 1) ≡ 0 (mod 3), and λu(u 1)(g 2 h 2 ) ≡ 0 (mod 12). These necessary conditions are shown to be sufficient for all λ≥ 2. The known existence result for λ = 1 is also improved.     

Super-simple group divisible designs with block size 4 and index 5

In this paper, we investigate the existence of a super-simple (4, 5)-GDD of type g^u and show that such a design exists if and only if u≥4, g(u−2)≥10, and .     

Lifting of divisible designs

The aim of this paper is to present a construction of t-divisible designs (DDs) for t > 3, because such DDs seem to be missing in the literature. To this end, tools such as finite projective spaces and their algebraic varieties are employed. More precisely, in a first step an abstract construction, called t-lifting, is developed. It starts from a set X containing a t-DD and a group G acting on X. Then several explicit examples are given, where X is a subset of PG(n,q) and G is a subgroup of GL_n + 1(q). In some cases X is obtained from a cone with a Veronesean or an h-sphere as its basis. In other examples, X arises from a projective embedding of a Witt design. As a result, for any integer t ≥ 2 infinitely many non-isomorphic t-DDs are found.Dedicated to Walter Benz on the occasion of his 75th birthday.