Some new large sets of geometric designs of type

Let V be an n-dimensional vector space over Fq. By a geometric t-[q, k, λ] design we mean a collection D of k-dimensional subspaces of V , called blocks, such that every t-dimensional subspace T of V appears in exactly λ blocks in D. A large set, LS[N][t, k, q], of geometric designs, is a collection of N t-[q, k, λ] designs which partitions the collection [ V k ] of all k-dimensional subspaces of V . Prior to recent article [4] only large sets of geometric 1-designs were known to exist. However in [4] M. Braun, A. Kohnert, P. Östergard, and A. Wasserman constructed the world’s first large set of geometric 2-designs, namely an LS[3][2,3,2], invariant under a Singer subgroup in GL8(2). In this work we construct an additional 9 distinct, large sets LS[3][2,3,2], with the help of lattice basis-reduction. 2010 MSC: 05B25, 05B40, 05E18