Algorithms for computing network coding rate regions via single element extensions of matroids

We propose algorithms for finding extreme rays of rate regions achievable with vector linear codes over finite fields F_q, q ∈ {2, 3, 4} for which there are known forbidden minors for matroid representability. We use the idea of single element extensions (SEEs) of matroids and enumeration of non-isomorphic matroids using SEEs, to first propose an algorithm to obtain lists of all non-isomorphic matroids representable over a given finite field.We modify this algorithm to produce only the list of all non-isomorphic connected matroids representable over the given finite field. We then integrate the process of testing which matroids in a list of matroids form valid linear network codes for a given network within matroid enumeration. We name this algorithm, which essentially builds all matroids that form valid network codes for a given network from scratch, as network-constrained matroid enumeration.