Gray codes for vector spaces are considered in two graphs: the
Grassmann graph, and the projective-space graph, both of which have
recently found applications in network coding. For the Grassmann
graph, constructions of cyclic optimal codes are given for all
parameters. As for the projective-space graph, two constructions for
specific parameters are provided, as well some non-existence results.

Furthermore, encoding and decoding algorithms are given for the
Grassmannian Gray code, which induce an enumerative-coding scheme. The
computational complexity of the algorithms is at least as low as known
schemes, and for certain parameter ranges, the new scheme outperforms
previously-known ones.