A matroid theory approach to multicast network coding

Network coding encourages the mixing of information flows at intermediate nodes of a network for enhanced network capacity, especially for one-to-many multicast applications. A fundamental problem in multicast network coding is to construct a feasible solution such that encoding and decoding are performed over a finite field of size as small as possible. Coding operations over very small finite fields (e.g., F₂) enable low computational complexity in theory and ease of implementation in practice. In this work, we propose a new approach based on matroid theory to study multicast network coding and its minimum field size requirements. Applying this new approach that translates multicast networks into matroids, we derive the first upper-bounds on the field size requirement based on the number of relay nodes in the network, and make new progresses along the direction of proving that coding over very small fields (F₂ and F₃) suffices for multicast network coding in planar networks.