  A $t$-$(n,k,\lambda)$ design over $\F_q$ is a collection of $k$-dimensional
subspaces of $\F_q^n$, called blocks, such that each $t$-dimensional subspace
of $\F_q^n$ is contained in exactly $\lambda$ blocks. Such $t$-designs over
$\F_q$ are the $q$-analogs of conventional combinatorial designs. Nontrivial
$t$-$(n,k,\lambda)$ designs over $\F_q$ are currently known to exist only for
$t \leq 3$. Herein, we prove that simple (meaning, without repeated blocks)
nontrivial $t$-$(n,k,\lambda)$ designs over $\F_q$ 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.
