Nontrivial t-designs over finite fields exist for all t

A t-(n,k,λ)t-(n,k,λ) design over FqFq is a collection of k  -dimensional subspaces of Fqn, called blocks, such that each t  -dimensional subspace of Fqn is contained in exactly λ blocks. Such t  -designs over FqFq are the q  -analogs of conventional combinatorial designs. Nontrivial t-(n,k,λ)t-(n,k,λ) designs over FqFq are currently known to exist only for t⩽3t⩽3. Herein, we prove that simple (meaning, without repeated blocks) nontrivial t-(n,k,λ)t-(n,k,λ) designs over FqFq exist for all t and q  , provided that k>12(t+1)k>12(t+1) and n is sufficiently large. This may be regarded as a q-analog of the celebrated Teirlinck theorem for combinatorial designs.