Network Coding Solutions for the Combination Network and its Subgraphs

The combination network is one of the simplest and insightful networks in coding theory. The vector network coding solutions for this network and some of its sub-networks are examined. For a fixed alphabet size of a vector network coding solution, an upper bound on the number of nodes in the network is obtained. This bound is an MDS bound for subspaces over a finite field. A family of sub-networks of combination networks is defined. It is proved that for this family of networks, which are minimal multicast networks, there is a gap in the minimum alphabet size between vector network coding solutions and scalar network coding solutions. This gap is obtained for any number of messages and is based on coloring of the q-Kneser graph and a new hypergraph generalization for it.