Large sets of t-designs over finite fields exist for all t

Abstract

A t-$(n,k,\lambda )_q$ design is a set of k-dimensional subspaces, called blocks, of an n-dimensional vector space V over the finite field $\mathbb {F}_q$ with q elements such that each t-dimensional subspace is contained in exactly $\lambda$ blocks. A partition of the complete set of k-dimensional subspaces of V into l disjoint t-$(n, k, \lambda )_q$ designs is called a large set of t-designs over $\mathbb {F}_q$ and denoted by $LS_q[l](t,k,n)$. Large sets of t-designs over finite fields with $t\ge 2$ are currently known to exist only for $LS_2[3](2,3,8)$, $LS_3[2](2,k,n)$ and $LS_5[2](2,k,n)$ with integers $n\ge 6, n\equiv 2\pmod 4$ and $3\le k\le n-3, k \equiv 3 \pmod 4$. In this article, we use the KLP theorem to prove that an $LS_q[l](t,k,n)$ exists for all t and q, provided that $k > 6.1(t+1)$ and n is sufficiently large and $\begin{bmatrix} n-s \\ k-s \end{bmatrix}_q\equiv 0 \pmod l\ \mathrm{for}\ 0\le s\le t.$