  Grassmannian codes are known to be useful in error-correction for random
network coding. Recently, they were used to prove that vector network codes
outperform scalar linear network codes, on multicast networks, with respect to
the alphabet size. The multicast networks which were used for this purpose are
generalized combination networks. In both the scalar and the vector network
coding solutions, the subspace distance is used as the distance measure for the
codes which solve the network coding problem in the generalized combination
networks. In this work we show that the subspace distance can be replaced with
two other possible distance measures which generalize the subspace distance.
These two distance measures are shown to be equivalent under an orthogonal
transformation. It is proved that the Grassmannian codes with the new distance
measures generalize the Grassmannian codes with the subspace distance and the
subspace designs with the strength of the design. Furthermore, optimal
Grassmannian codes with the new distance measureshave minimal requirements for
network coding solutions of some generalized combination networks. The coding
problems related to these two distance measures, especially with respect to
network coding, are discussed. Finally, by using these new concepts it is
proved that codes in the Hamming scheme form a subfamily of the Grassmannian
codes.
