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.
Abstract-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.
This paper investigates the enumeration, rate region computation, and hierarchy of general multi-source multi-sink hyperedge networks under network coding, which includes multiple network models, such as independent distributed storage systems and index coding problems, as special cases. A notion of minimal networks and a notion of network equivalence under group action are defined. An efficient algorithm capable of directly listing single minimal canonical representatives from each network equivalence class is presented and utilized to list all minimal canonical networks with up to 5 sources and hyperedges. Computational tools are then applied to obtain the rate regions of all of these canonical networks, providing exact expressions for 744,119 newly solved network coding rate regions corresponding to more than 2 trillion isomorphic network coding problems. In order to better understand and analyze the huge repository of rate regions through hierarchy, several embedding and combination operations are defined so that the rate region of the network after operation can be derived from the rate regions of networks involved in the operation.The embedding operations enable the definition and determination of a list of forbidden network minors for the sufficiency of classes of linear codes. The combination operations enable the rate regions of some larger networks to be obtained as the combination of the rate regions of smaller networks. The integration of both the combinations and embedding operators is then shown to enable the calculation of rate regions for many networks not reachable via combination operations alone.
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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