  We study scalar-linear and vector-linear solutions of the generalized
combination network. We derive new upper and lower bounds on the maximum number
of nodes in the middle layer, depending on the network parameters and the
alphabet size. These bounds improve and extend the parameter range of known
bounds. Using these new bounds we present a lower bound and an upper bound on
the gap in the alphabet size between optimal scalar-linear and optimal
vector-linear network coding solutions. For a fixed network structure, while
varying the number of middle-layer nodes $r$, the asymptotic behavior of the
upper and lower bounds shows that the gap is in $\Theta(\log(r))$.
