Network Generalized Hamming Weight

In this paper, we extend the notion of generalized Hamming weight for classical linear block code to linear network codes by introducing the network generalized Hamming weight (NGHW) of a linear block code with respect to a given linear network code. The basic properties of NGHW are studied. We further show that NGHW can be used as a tool to characterize the security performance of a linear network code on a wiretap network. We also introduce the idea of Network Maximum Distance Separation code (NMDS code) by extending the notion of Maximum Distance Separation code in classical algebraic coding theory. We prove that NMDS codes play an important role in minimizing the information that an eavesdropper can obtain from the network.