有限域上的非平凡t-设计对所有t都存在
Nontrivial t-Designs over Finite Fields Exist for All t
摘要：
基于$\F_q$的$t$ - $(n,k,λ)$设计是$\F_q^n$的$k$维子空间（称为块）的集合，使得$\F_q^n$的每个$t$维子空间恰好包含在$λ$块中.这种优于$\F_q$的$t$ -设计是常规组合设计的$q$ -类似物.非平凡$t$ -$\F_q$上的$(n,k,λ)$设计目前已知仅存在于$t \leq 3$中。在这里，我们证明了简单的（意思是，没有重复的块）非平凡的$t$ - $(n,k,λ)$设计在$\F_q$上存在所有的$t$和$q$，只要$k > 12t$和$n$是足够大的。这可以看作是一个$q$ -模拟著名的Teirlinck定理的组合设计。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
