The Role of the Alphabet in Network Coding: An Optimization Approach

We consider the problem of determining the one-shot, zero-error capacity of a coded, multicast network over a small alphabet. We introduce a novel approach to this problem based on a mixed-integer program, which computes the size of the largest unambiguous codebook for a given alphabet size. As an application of our approach, we recover, extend and refine various results that were previously obtained with case-by-case analyses or specialized arguments, giving evidence of the wide applicability of our approach. We also provide two simple ideas that reduce the complexity of our method for some families of networks. We conclude the paper by outlining a research program we wish to pursue to investigate the one-shot capacity of large networks affected by adversarial noise and, more generally, the role played by the alphabet size in network coding.