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 $\Theta(\log(r))$.
- Pub Date:
- June 2020
- Computer Science - Information Theory;
- Computer Science - Networking and Internet Architecture;
- Computer Science - Social and Information Networks;
- Mathematics - Combinatorics
- extended version of arXiv:2001.04150v2