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 Fqn, called blocks, such that each t-dimensional subspace of Fqn 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⩽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>12(t+1) and n is sufficiently large. This may be regarded as a q-analog of the celebrated Teirlinck theorem for combinatorial designs.