An explicit construction for large sets of infinite dimensional q $q$‐Steiner systems.

Let V $V$ be a vector space over the finite field Fq ${{\mathbb{F}}}_{q}$. A q $q$‐Steiner system, or an S(t,k,V)q $S{(t,k,V)}_{q}$, is a collection ℬ ${\rm{{\mathcal B}}}$ of k $k$‐dimensional subspaces of V $V$ such that every t $t$‐dimensional subspace of V $V$ is contained in a unique element of ℬ ${\rm{{\mathcal B}}}$. A large set of q $q$‐Steiner systems, or an LS(t,k,V)q $LS{(t,k,V)}_{q}$, is a partition of the k $k$‐dimensional subspaces of V $V$ into S(t,k,V)q $S{(t,k,V)}_{q}$ systems. In the case that V $V$ has infinite dimension, the existence of an LS(t,k,V)q $LS{(t,k,V)}_{q}$ for all finite t,k $t,k$ with 1<t<k $1\lt t\lt k$ was shown abstractly by Cameron in 1995. This paper provides an explicit construction of an LS(t,t+1,V)q $LS{(t,t+1,V)}_{q}$ for all prime powers q $q$, all positive integers t $t$, and where V $V$ has countably infinite dimension. [ABSTRACT FROM AUTHOR]