Network combination operations preserving the sufficiency of linear network codes

Abstract

Operations that combine smaller networks into a larger network in a manner such that the rate region of the larger network can be directly obtained from the associated rate regions of the smaller networks are defined and presented. The operations are selected to also have the property that the sufficiency of classes of linear network codes and the tightness of the Shannon (polymatroid) outer bound are preserved in the combination. Four such operations are defined, and the classes of linear codes considered include both scalar and vector codes. It is demonstrated that these operations enable one to obtain rate regions for networks of arbitrary size, and to also determine if some classes of linear codes are sufficient and/or the Shannon outer bound bound is tight for these networks.