Large sets of subspace designs

To the memory of Axel Kohnert 1962–2013
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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 LS5[2](2,3,6). The application of the new recursion method to this halving and an already known LS3[2](2,3,6) yields two infinite two-parameter series of halvings LS3[2](2,k,v) and LS5[2](2,k,v) with integers v6, v2(mod4) and 3kv3, k3(mod4).

Thus 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.

Keywords

q-analog
Combinatorial design
Subspace design
Large set
Halving

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