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 $\Theta(\log(r))$.

翻译：我们研究通用组合网络的斜线和矢量线性解决方案。我们根据网络参数和字母大小,从中间层最大节点数中得出新的上下界限。这些界限改进并扩大了已知界限的参数范围。使用这些新界限,我们提出了一个较低的界限,在最佳斜线和最佳矢量线性网络编码解决方案之间字母大小的上下界限。对于固定的网络结构来说,在改变中层节点数的同时,上层和下层节点的无序行为表明,差距是$\Theta(\log(log(r))$。