Packing and covering properties of subspace codes

Codes in the projective space over a finite field, referred to as subspace codes, and in particular codes in the Grassmannians, referred to as constant-dimension codes (CDCs), have been proposed for error control in random network coding. In this paper, we study the packing and covering properties of subspace codes, which can be used with the subspace metric or the injection metric. We first determine some fundamental geometric properties of the projective space. Using these results, we derive bounds on the cardinalities of packing and covering subspace codes, and determine the asymptotic rate of optimal packing and optimal covering subspace codes for both metrics. We thus show that optimal packing CDCs are asymptotically optimal packing subspace codes for both metrics. However, optimal covering CDCs can be used to construct asymptotically optimal covering subspace codes only for the injection metric.