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Dualities Between Entropy Functions and Network Codes

机译：熵函数与网络代码之间的对偶

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摘要

In communications networks, the capacity region of multisource network coding is given in terms of the set of entropy functions Gamma*. More broadly, determination of Gamma* would have an impact on converse theorems for multi-terminal problems in information theory. This paper provides several new dualities between entropy functions and network codes. Given a function g ges 0 defined on all subsets of N random variables, we provide a construction for a network multicast problem which is ldquosolvablerdquo if and only if g is the entropy function of a set of quasi-uniform random variables. The underlying network topology is fixed and the multicast problem depends on g only through link capacities and source rates. A corresponding duality is developed for linear network codes, where the constructed multicast problem is linearly solvable if and only if g is linear group characterizable. Relaxing the requirement that the domain of g be subsets of random variables, we obtain a similar duality between polymatroids and the linear programming bound. These duality results provide an alternative proof of the insufficiency of linear (and abelian) network codes, and demonstrate the utility of non-Shannon inequalities to tighten outer bounds on network coding capacity regions.
机译：在通信网络中，多源网络编码的容量区域是根据熵函数Gamma * 的集合给出的。更广泛地说，Gamma * 的确定将对信息论中多终端问题的逆定理产生影响。本文提供了熵函数和网络代码之间的几个新对偶。给定在 N 个随机变量的所有子集上定义的函数 g ges 0，我们提供了一个网络多播问题的构造，当且仅当 g < / i>是一组准均匀随机变量的熵函数。底层网络拓扑是固定的，多播问题仅取决于链路容量和源速率取决于 g 。针对线性网络代码开发了相应的对偶性，其中，且仅当 g 具有线性组可表征性时，构造的多播问题才可线性解决。放宽 g 的域是随机变量子集的要求，我们在多拟阵和线性规划边界之间获得了相似的对偶性。这些对偶结果为线性（和阿贝尔）网络代码不足提供了另一种证明，并证明了非香农不等式的用途是加强网络编码能力区域的外部界限。

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