New Geometric Large Sets

Let V be an n-dimensional vector space over the field of q elements. 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 subspace T of V appears in exactly λ blocks in D. A large set, LS [N] [t, k, q^n], of geometric designs is a collection on N disjoint t-[q^n, k, λ] designs that partitions [V K], the collection of k-dimensional subspaces of V. In this work we construct non-isomorphic large sets using methods based on incidence structures known as the Kramer-Mesner matrices. These structures are induced by particular group actions on the collection of subspaces of the vector space V. Subsequently, we discuss and use computational techniques for solving certain linear problems of the form AX = B, where A is a large integral matrix and X is a {0,1} solution. These techniques involve (i) lattice basis-reduction, including variants of the LLL algorithm, and (ii) linear programming. Inspiration came from the 2013 work of Braun, Kohnert, Ostergard, and Wassermann, [17], who produced the first nontrivial large set of geometric designs with t ≥ 2. Bal Khadka and Michael Epstein provided the know-how for using the LLL and linear programming algorithms that we implemented to construct the large sets.