Lifting of divisible designs

Abstract

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

Divisible designs, Laguerre geometry, and beyond

“…A generalisation of Spera's construction was exhibited in [54] and [56]. It was put into a more general context in [38] as follows: Let a group G acting on some set X and a starter t-DD in X be given. Then, under certain technical conditions, a new t-DD can be obtained via the action of G on X.…”

Divisible designs with dual translation group

“…DDs gained, e.g., by a well known construction of A.G. Spera which is used in [13], or by a construction, called construction (A), given in [6], or by a generalization of both presented in [2], admit elementary abelian dual translation groups. As the main result, Theorem 3.1 shows that a DD which possesses such an automorphism group can be characterized as being isomorphic to a substructure of a finite affine space.…”