Nontrivial t-designs over finite fields exist for all t

A t-(n, k, λ) design over Fq is a collection of k-dimensional subspaces of Fq , called blocks, such that each t-dimensional subspace of Fq is contained in exactly λ blocks. Such t-designs over Fq are the q-analogs of conventional combinatorial designs. Nontrivial t-(n, k, λ) designs over Fq are currently known to exist only for t 6 3. Herein, we prove that simple (meaning, without repeated blocks) nontrivial t-(n, k, λ) designs over Fq exist for all t and q, provided that k > 12t and n is sufficiently large. This may be regarded as a q-analog of the celebrated Teirlinck theorem for combinatorial designs. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 126 (2013)