This paper provides new dualities be-linear network codes. We show that the constructed multween entropy functions and network codes. These ticast problem is linearly solvable if and only if h is an duality results give an alternative proof of the insufentropy function for random variables generated by vecficiency of linear (and abelian) network codes, and tor spaces. A corresponding limiting form of this duality demonstrate the utility of non-Shannon inequalities is also provided. Finally, by relaxing the requirement that to tighten outer bounds on network coding capacity the domain of h be subsets of random variables, we obregions. tain in Theorem 3 a duality between polymatroids and the linear programming bound. These dualities resolve I. INTRODUCTION the open problem from [14], where a special case of Theorem 1 was proved for four variables. One fundamental problem in network coding is to un-These duality results yield several immediate corollarderstand the capacity region and the classes of codes that ies. In particular, we provide an alternative proof to [4,15] achieve capacity. In the single session multicast scenario, for the insufficiency of linear (and abelian) network codes the problem is well understood. In particular, the capac-for the multi-source problem, and demonstrate the utility ity region is characterized by max-flow/min-cut bounds of non-Shannon inequalities to tighten outer bounds on and linear network codes are sufficient to achieve maximal network coding capacity regions. throughput [1, 4]. Significant practical and theoretical complications
It was shown in REF that fractional linear network codes over finite rings (and modules) are special cases of codes generated by Abelian groups.
Dualities between Entropy Functions and Network Codes
2008 Fourth Workshop on Network Coding, Theory and Applications
2008 Fourth Workshop on Network Coding, Theory and Applications
Computer Science Mathematics