Refined Coding Bounds and Code Constructions for Coherent Network Error Correction

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.Comment: 32 page

Network Coding for Error Correction

In this thesis, network error correction is considered from both theoretical and practical viewpoints. Theoretical parameters such as network structure and type of connection (multicast vs. nonmulticast) have a profound effect on network error correction capability. This work is also dictated by the practical network issues that arise in wireless ad-hoc networks, networks with limited computational power (e.g., sensor networks) and real-time data streaming systems (e.g., video/audio conferencing or media streaming). Firstly, multicast network scenarios with probabilistic error and erasure occurrence are considered. In particular, it is shown that in networks with both random packet erasures and errors, increasing the relative occurrence of erasures compared to errors favors network coding over forwarding at network nodes, and vice versa. Also, fountain-like error-correcting codes, for which redundancy is incrementally added until decoding succeeds, are constructed. These codes are appropriate for use in scenarios where the upper bound on the number of errors is unknown a priori. Secondly, network error correction in multisource multicast and nonmulticast network scenarios is discussed. Capacity regions for multisource multicast network error correction with both known and unknown topologies (coherent and noncoherent network coding) are derived. Several approaches to lower- and upper-bounding error-correction capacity regions of general nonmulticast networks are given. For 3-layer two-sink and nested-demand nonmulticast network topologies some of the given lower and upper bounds match. For these network topologies, code constructions that employ only intrasession coding are designed. These designs can be applied to streaming erasure correction code constructions.</p