In this work, we extend the notion of generalized Hamming weight for classical linear block codes to linear network codes by introducing the network generalized Hamming weight (NGHW) for a given network with respect to a fixed 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 notion of network maximum distance separable code (NMDS code) by extending the notion of Maximum Distance Separable 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.

Coherent network error correction is the error-control problem in network coding with the knowledge of the network codes at the source and sink nodes. With respect to a given set of local encoding kernels defining a linear network code, we obtain refined versions of the Hamming bound, the Singleton bound, and the Gilbert-Varshamov bound for coherent network error correction. Similar to its classical counterpart, this refined Singleton bound is tight for linear network codes. The tightness of this refined bound is shown by two construction algorithms of linear network codes achieving this bound. These two algorithms illustrate different design methods: one makes use of existing network coding algorithms for error-free transmission and the other makes use of classical error-correcting codes. The implication of the tightness of the refined Singleton bound is that the sink nodes with higher maximum flow values can have higher error correction capabilities.

In this paper, we introduce a network multicast problem called the alternating multicast problem. In this problem, there is more than one multicast on the network, but only one of the multicasts can take place at any time. We propose an algorithm for constructing a linear network code which is static in the sense that the coding operations do not depend on the multicast. The code so constructed achieves the max-flow bound for each multicast, i.e., optimality is achieved when the base field is sufficiently large. An upper bound on the required field size for constructing the code is also obtained. We conjecture that a network code constructed by our algorithm can be converted into a network code for the associated simultaneous multicast problem, i.e., all the multicasts can take place at the same time. An example is presented to illustrate our conjecture.

In this paper, we present a refined version of the Singleton bound for network error correction, and propose an algorithm for constructing network codes that achieve this bound.

Network coding theory shows that the multicast rate in a network can be increased if coding is allowed in the network nodes. Thus the capacity of a network with network coding is generally larger than that with routing alone. We quantify this gain in closed form for a class of networks called combination networks. From this result, it can readily be deduced that network coding gain can be unbounded.