Nontrivial t-Designs over Finite Fields Exist for All t

Understand

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.