2. q-analogs of group divisible designs

Abstract

A well known class of objects in combinatorial design theory are group divisible designs. Here, we introduce the q-analogs of group divisible designs. It turns out that there are interesting connections to scattered subspaces, q-Steiner systems, packing designs and q r -divisible projective sets.We give necessary conditions for the existence of q-analogs of group divisible designs, construct an infinite series of examples, and provide further existence results with the help of a computer search.One example is a (6, 2, 3, 2)2 group divisible design over GF(2) which is a packing design consisting of 180 blocks that such every 2-dimensional subspace in GF(2) 6 is covered at most twice.

Subspace packings: constructions and bounds

“…We remark that A 2 (6, 3, 2; 4) ≥ 360, which was obtained in the context of q-GDDs [14], was also obtained in [24]. The upper bound for A 2 (6, 4, 2; 2), based on integer linear programming, need a more detailed explanation, which is marked by a ⋆ in the corresponding table.…”

“…For designs the known necessary existence criteria also have their q-analog counterparts. Interestingly enough, for group divisible designs there is an additional necessary existence criterion for q > 1, see [14]. Also the Johnson bound for constant-dimension codes, see Proposition 2 for λ = 1, was improved [57].…”

“…We will adjust this method in Subsection 4.2. Some tailored constructions that indeed meet the known upper bounds are stated in Subsection 4.3. q-analogs of group divisible designs also give some good constructions for a few parameters, see [14]. Of course a packing design is the best that can be achieved, so that we also refer to the corresponding literature, see e.g.…”

“…Various q-analogs of designs were considered, t-designs (see [11] and references therein), Steiner systems [9,23] and in particular the Fano plane [24,58], transversal designs [27], group divisible designs [14], large sets [12,13], etc. But, one very natural modification of the design property was not thoroughly studied -the family of packings.…”

A subspace code of size <inline-formula><tex-math id="M1">\begin{document}$ \bf{333} $\end{document}</tex-math></inline-formula> in the setting of a binary <inline-formula><tex-math id="M2">\begin{document}$ \bf{q} $\end{document}</tex-math></inline-formula>-analog of the Fano plane

“…A t-(v, k, λ) q packing design is a set of k-dimensional subspaces of F v q such that every t-dimensional subspace is covered at most λ times. The 2-(6, 3, 2) 2 packing design of cardinality 180 with an automorphism group of order 9 from [11] was quickly rediscovered using the presented algorithmic approach. The packing design is indeed optimal, which can be shown using a Johnson-type argument.…”