Some new large sets of geometric designs of type LS[3][2; 3; 2^8]

Let V be an n-dimensional vector space over F q. By a geometric t-[q^n; k; λ] design we mean a collection D of k-dimensional subspaces of V , called blocks, such that every t-dimensional of V appears in exactly blocks in D: A large set, LS[N][t; k; q^n], of geometric designs, is a collection ofsubspace T N t-[qn; 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][2,3,2^8], invariant under a Singer subgroup in GL8(2). In this work we construct an additional 9 distinct, large sets LS[3][2,3,2^8], with the help of lattice basis-reduction.