Nontrivial t-Designs over Finite Fields Exist for All t

A $t$-$(n,k,λ)$ 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 $λ$ blocks. Such $t$-designs over $\F_q$ are the $q$-analogs of conventional combinatorial designs. Nontrivial $t$-$(n,k,λ)$ designs over $\F_q$ are currently known to exist only for $t \leq 3$. Herein, we prove that simple (meaning, without repeated blocks) nontrivial $t$-$(n,k,λ)$ 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.

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