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

TL;DR

This paper proves that determining the existence of a network coding scheme with some fixed-size messages and edges is undecidable, extending the understanding of the computational limits of network coding problems.

Contribution

It demonstrates the undecidability of network coding with fixed-size components, providing a partial solution to the broader conjecture about undecidability in network coding.

Findings

Decidability is impossible for certain fixed-size network coding configurations.

The proof uses analogies with digital circuits and Turing machine simulation.

This work extends undecidability results to more constrained network coding scenarios.

Abstract

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