Q-Analogs of Designs: Subspace Designs

Abstract

For discrete structures which are based on a finite ambient set and its subsets there exists the notion of a “q-analog”: For this, the ambient set is replaced by a finite vector space and the subsets are replaced by subspaces. Consequently, cardinalities of subsets become dimensions of subspaces. Subspace designs are the q-analogs of combinatorial designs. Introduced in the 1970s, these structures gained a lot of interest recently because of their application to random network coding. In this chapter we give a thorough introduction to the subject starting from the subspace lattice and its automorphisms, the Gaussian binomial coefficient and counting arguments in the subspace lattice. This prepares us to survey the known structural and existence results about subspace designs. Further topics are the derivation of subspace designs with related parameters from known subspace designs, as well as infinite families, intersection numbers, and automorphisms of subspace designs. In addition, q-Steiner systems and so called large sets of subspace designs will be covered. Finally, this survey aims to be a comprehensive source for all presently known subspace designs and large sets of subspace designs with small parameters.