Packing and Covering Properties of CDCs and Subspace Codes

Codes in the projective space over a finite field, referred to as subspace codes, and in particular codes in the Grassmannian, referred to as constant-dimension codes (CDCs), have been proposed for error control in random network coding. In this paper, we first study the covering properties of CDCs. We determine some fundamental geometric properties of the Grassmannian. Using these properties, we derive bounds on the minimum cardinality of a CDC with a given covering radius and determine the asymptotic rate of optimal covering CDCs. We then study the packing and covering properties of subspace codes, which can be used with the subspace metric or the modified subspace metric. We investigate the properties of balls in the projective space. Using these results, we derive bounds on the cardinalities of packing and covering subspace codes, and we 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 modified subspace metric.