  This paper considers vector network coding solutions based on rank-metric
codes and subspace codes. The main result of this paper is that vector
solutions can significantly reduce the required alphabet size compared to the
optimal scalar linear solution for the same multicast network. The multicast
networks considered in this paper have one source with $h$ messages, and the
vector solution is over a field of size $q$ with vectors of length~$t$. For a
given network, let the smallest field size for which the network has a scalar
linear solution be $q_s$, then the gap in the alphabet size between the vector
solution and the scalar linear solution is defined to be $q_s-q^t$. In this
contribution, the achieved gap is $q^{(h-2)t^2/h + o(t)}$ for any $q \geq 2$
and any even $h \geq 4$. If $h \geq 5$ is odd, then the achieved gap of the
alphabet size is $q^{(h-3)t^2/(h-1) + o(t)}$. Previously, only a gap of size
size one had been shown for networks with a very large number of messages.
These results imply the same gap of the alphabet size between the optimal
scalar linear and some scalar nonlinear network coding solution for multicast
networks. For three messages, we also show an advantage of vector network
coding, while for two messages the problem remains open. Several networks are
considered, all of them are generalizations and modifications of the well-known
combination networks. The vector network codes that are used as solutions for
those networks are based on subspace codes, particularly subspace codes
obtained from rank-metric codes. Some of these codes form a new family of
subspace codes, which poses a new research problem.
