Nontrivial t-Designs over Finite Fields Exist for All t

TL;DR

This paper proves the existence of nontrivial t-designs over finite fields for all t and q, given certain parameters, extending the known existence range significantly.

Contribution

It establishes the existence of simple nontrivial t-designs over finite fields for all t and q under specific conditions, generalizing previous results.

Findings

Existence of t-designs for all t and q proven.

Designs exist when k > 12t and n is large enough.

Extends the theory of q-analogs of combinatorial designs.

Abstract

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.