On Network Matroids And Linear Network Codes

This paper deals with matroids on the edge set of a network. Through the structure of edge-disjoint paths, a single-source network is associated with a network matroid, which turns out to be representable. A linear network code on an acyclic network assigns a coding vector to every edge. The linear independence among coding vectors naturally induces a matroid. It is shown that every independent set in the matroid so induced is also independent in the network matroid. Moreover, the two matroids coincide with each other if and only if the linear network code is a generic one. Furthermore, every representation for the network matroid of an acyclic network induces a generic linear network code. This offers a new characterization of generic linear network codes. For the network matroid of a cyclic network, an algorithm for finding a representation is also derived through the association with an acyclic network.