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

Abstract

Let be a vector space over the finite field . A ‐Steiner system, or an , is a collection of ‐dimensional subspaces of such that every ‐dimensional subspace of is contained in a unique element of . A large set of ‐Steiner systems, or an , is a partition of the ‐dimensional subspaces of into systems. In the case that has infinite dimension, the existence of an for all finite with was shown abstractly by Cameron in 1995. This paper provides an explicit construction of an for all prime powers , all positive integers , and where has countably infinite dimension.