Capacity regions for linear and abelian network codes

While linear network codes are proved suboptimal in general multicast scenarios, the loss of throughput due to the use of linear network codes is still unknown. This paper attempts to investigate the loss in throughput by identifying the capacity regions for linear network codes. We prove that the capacity region can be identified by taking intersection of a set of hyperplanes induced by the network and the convex cone closure of the set of all linear representable entropy functions. We also extend the study of network coding capacity region to abelian network codes which contain linear network codes as a subclass. For the case of two multicast sessions, we obtain an inner bound making use of the convex closure of entropy functions which are abelian group representable.