  Constant-dimension codes have recently received attention due to their
significance to error control in noncoherent random linear network coding. What
the maximal cardinality of any constant-dimension code with finite dimension
and minimum distance is and how to construct the optimal constant-dimension
code (or codes) that achieves the maximal cardinality both remain open research
problems. In this paper, we introduce a new approach to solving these two
problems. We first establish a connection between constant-rank codes and
constant-dimension codes. Via this connection, we show that optimal
constant-dimension codes correspond to optimal constant-rank codes over
matrices with sufficiently many rows. As such, the two aforementioned problems
are equivalent to determining the maximum cardinality of constant-rank codes
and to constructing optimal constant-rank codes, respectively. To this end, we
then derive bounds on the maximum cardinality of a constant-rank code with a
given minimum rank distance, propose explicit constructions of optimal or
asymptotically optimal constant-rank codes, and establish asymptotic bounds on
the maximum rate of a constant-rank code.
