Subspace Codes for Random Networks Based on Plücker Coordinates and Schubert Cells

The construction of codes in the projective space for error control in random networks has been the focus of recent research. The Plücker coordinate description of subspaces has been discussed in the context of constant dimension codes, as well as the Schubert cell description of certain code parameters. In this paper we use this classical tool to reformulate some standard constructions of constant dimension codes and give a unified framework. We present a general method of constructing non-constant dimension subspace codes with respect to minimum subspace distance or minimum injection distance as the union of constant dimension subspace codes restricted to selected Schubert cells. The selection of these Schubert cells is based on the subset distance of tuples corresponding to the Plücker coordinate matrices associated with the subspaces contained in the respective Schubert cells. In this context, we show that a recent construction of non-constant dimension Ferrers-diagram rank-metric subspace codes is subsumed in our framework.