On q-analogs of group divisible designs

Titelangaben

de Beule, Jan ; Kiermaier, Michael ; Kurz, Sascha ; Wassermann, Alfred:
On q-analogs of group divisible designs.
2019
Veranstaltung: 9th Slovenian International Conference on Graph Theory , 23.-29.06.2019 , Bled, Slovenia.
(Veranstaltungsbeitrag: Kongress/Konferenz/Symposium/Tagung , Vortrag )

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Abstract

Group divsible designs are well-studied objects in combinatorics. Recently, q-analogs of group divisible designs (q-GDDs) have been introduced. Let v, g, k, and
λ be sets of positive integers and let be a positive integer. The q-analog of a group divisible design of index
λ and order v with parameters (v,g,k,λ)_q is a triple
(V,G,V), where V is a vector space of dimension v over GF(q), G is a partition of V into g -dimensional subspaces (groups), and V is a family of k-dimensional subspaces (blocks) of V such that every 2-dimensional subspace of V
occurs in exactly λ blocks or one group, but not both.
q-analogs of group divisible designs are connected to scattered subspaces in finite geometry, q-analogs of Steiner systems, subspace design packings and more.
After an introduction to the subject, recent results are presented with special attention on cases where the vector space partition is a non-Desarguesian spread. For example, there is no (8,4,4,7)_2 GDD and a (8,4,4,14)_2 GDD exists only for the Desarguesian spread.