Large sets of subspace designs

Abstract

Abstract. In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Graßmannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs.We construct a 2-(6, 3, 78) 5 design by computer, which corresponds to a halving LS 5 [2](2, 3, 6). The application of the new recursion method to this halving and an already known LS 3 [2](2, 3, 6) yields two infinite two-parameter series of halvings LS 3 [2](2, k, v) and LS 5 [2](2, k, v) with integers v ≥ 6, v ≡ 2 (mod 4) and 3Thus in particular, two new infinite series of nontrivial subspace designs with t = 2 are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with t = 2.

Subspace packings: constructions and bounds

“…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.…”

An Assmus--Mattson Theorem for Rank Metric Codes

“…We list in Table 1 the known algebraic constructions of infinite families of t-(n, r, λ) subspace designs that do not rely on the existence of other subspace designs. This list is still rather short, although many specific numerical parameter sets are known to be realizable [8,9,10]. In most cases a computer search is combined with an assumed automorphism group, often in fact the normalizer of a Singer cycle group.…”

“…As an example, one of Suzuki's 2-n, 3, 3 1 q designs over F q yields a 2-n, 3, q 3 n − 5 1 q subspace design, whose Table 5]. Finally, in [8], the authors show how to construct new large sets of designs from existing ones. An LS q [N ](t, r, n) large set is a partition of the set of n r q r-dimensional…”

“…In [9, Table 7] the authors list 5 large set parameters that are known to be realizable, and many more that are admissable but for which existence is not yet established. The interested reader is referred to [8] for further details.…”

Disjoint q -Steiner systems in dimension 13

“…Since then, more subspace designs have been constructed, see [3,4,5,11,12,16,17,18]. In [2] the first q-Steiner systems for t = 2 were constructed.…”