A new computational technique is presented for determining rate regions for coded networks. The technique directly manipulates the extreme ray representation of inner and outer bounds for the region of entropic vectors. We use new inner bounds on region of entropic vectors based on conic hull of ranks of representable matroids. In particular, the extreme-ray representations of these inner bounds are obtained via matroid enumeration and minor exclusion. This is followed by a novel use of iterations of the double description method to obtain the desired rate regions. Applications in multilevel diversity coding systems (MDCS) are discussed as an example. The special structure of the problem that makes this technique inherently fast along with being scalable is also discussed. Our results demonstrate that for each of the 31 2-level 3-encoder and the 69 3-level 3-encoder MDCS configurations, if scalar linear codes (over any field) suffice to achieve the rate region, then in fact binary scalar linear codes suffice. For the 31 2-level 3-encoder cases where scalar codes are insufficient we demonstrate that vector linear codes suffice and provide some explicit constructions of these codes.

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We propose algorithms for finding extreme rays of rate regions achievable with vector linear codes over finite fields Fq, 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 nonisomorphic 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 networkconstrained matroid enumeration.Index Terms-Single emement extensions, rate regions of multi-source network coding problems, region of entropic vectors

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