Matroidal Characterization of Optimal Linear Network Codes over Cyclic Networks

For a linear network code (LNC), various senses of optimality are defined by linear independence among certain coding vectors. A generic LNC is optimal in an extreme sense. Over an acyclic network, there has been a characterization of a generic LNC by the coincidence between the matroid of linearly independent sets of coding vectors of the LNC and the network matroid, which is defined by the existence of appropriate edge-disjoint paths. It turns out that this characterization is still valid when the network contains cycles, despite the fact that it is not straightforward to extend theoretic results on LNCs over acyclic networks to cyclic ones. Meanwhile, the variable-rate property of a generic LNC on an acyclic network also extends to cyclic networks.