On Coding for Non-Multicast Networks ∗
We consider the issue of coding for non-multicast networks. For multicast networks, it is known that linear operations over a field no larger than the number of receivers are sufficient to achieve all feasible connections. In the case of nonmulticast networks, necessary and sufficient conditions are known, if we restrict ourselves to linear codes over a finite field [1]. However, no linearity sufficiency results exist for non-multicast networks. Indeed, [2] shows that linearity over a field is not sufficient in general. We present a coding theorem that provides necessary and sufficient conditions, in terms of receiver entropies, for an arbitrary set of connections to be achievable on any network. We conjecture that linearity is sufficient to satisfy the coding theorem, when linear operations are performed over vectors rather than scalars in a field. We illustrate the intuition of this conjecture with an example. This work is part of an ongoing cooperation with R. Koetter.