We study scalar-linear and vector-linear solutions to 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. These bounds improve and extend the parameter range of known bounds. Using these new bounds we present a general lower bound on the gap in the alphabet size between scalar-linear and vector-linear solutions.

中文翻译：

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广义组合网络标量解与向量解的差距

我们研究广义组合网络的标量线性和向量线性解决方案。我们根据网络参数推导出中间层最大节点数的新上限和下限。这些边界改进和扩展了已知边界的参数范围。使用这些新边界，我们提出了标量线性和向量线性解决方案之间字母表大小差距的一般下限。