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We consider a network coding setting where some of the messages and edgeshave fixed alphabet sizes, that do not change when we increase the commonalphabet size of the rest of the messages and edges. We prove that the problemof deciding whether such network admits a coding scheme is undecidable. Thiscan be considered as a partial solution to the conjecture that network coding(without fixed-size messages/edges) is undecidable. The proof, which makesheavy use of analogies with digital circuits, is essentially constructing adigital circuit of logic gates and flip-flops within a network coding modelthat is capable of simulating an arbitrary Turing machine.

The Undecidability of Network Coding with some Fixed-Size Messages and Edges