Creation Date: Jul 24, 2009
Published In: Jul 2009
Paper Type: Dissertation
School: University of TorontoAbstract:
Network coding has emerged as a new paradigm for communication in networks, allowing packets to be algebraically combined at internal nodes, rather than simply routed or replicated. The very nature of packet-mixing, however, makes the system highly sensitive to error propagation. Classical error correction approaches are therefore insufficient to solve the problem, which calls for novel techniques and insights.
The main portion of this work is devoted to the problem of error control assuming an adversarial or worst-case error model. We start by proposing a general coding theory for adversarial channels, whose aim is to characterize the correction capability of a code. We then specialize this theory to the cases of coherent and noncoherent network coding. For coherent network coding, we show that the correction capability is given by the rank metric, while for noncoherent network coding, it is given by a new metric, called the injection metric. For both cases, optimal or near-optimal coding schemes are proposed based on rank-metric codes. In addition, we show how existing decoding algorithms for rank-metric codes can be conveniently adapted to work over a network coding channel. We also present several speed improvements that make these algorithms the fastest known to date.
The second part of this work investigates a probabilistic error model. Upper and lower bounds on capacity are obtained for any channel parameters, and asymptotic expressions are provided in the limit of long packet length and/or large field size. A simple coding scheme is presented that achieves capacity in both limiting cases. The scheme has fairly low decoding complexity and a probability of failure that decreases exponentially both in the packet length and in the field size in bits. Extensions of the scheme are provided for several variations of the channel.
A final contribution of this work is to apply rank-metric codes to a closely related problem: securing a network coding system against an eavesdropper. We show that the maximum possible rate can be achieved with a coset coding scheme based on rank-metric codes. Unlike previous schemes, our scheme has the distinctive property of being universal: it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying network code. In addition, the scheme can be easily combined with a rank-metric-based error control scheme to provide both security and reliability.