Network Coding Theory Via Commutative Algebra

The fundamental result of linear network coding asserts the existence of an optimal code on an acyclic single-source multicast network when the symbol field is sufficiently large. The restriction to acyclic networks turns out to stem from the customary structure of the symbol alphabet as a field. Adopting data units belonging to a discrete valuation ring (DVR), that is, a PID with a unique maximal ideal, much of the network coding theory extends to cyclic single-source multicast networks. Convolutional network coding is the instance of DVR-based network coding when the DVR consists of rational power series over the symbol field. Meanwhile, a field can be regarded as a degenerate DVR since it is a PID with the maximal ideal 0. Thus the conventional field-based network coding theory becomes a degenerate version of the DVR-based theory. This paper also delves into the issue of constructing optimal network codes on cyclic networks. Inspired by matroid duality theory, a novel method is devised to take advantage of all existing acyclic algorithms for network code construction. It associates every cyclic network with a quadratically large acyclic network so that essentially every optimal code on the acyclic network directly induces one on the cyclic network.