Katherine Morrison Extended Research Statement.dvi

Reliable transmission of digital data is essential in today’s society given its role in technological advances such as transmission of satellite images, wireless communications, and data storage/retrieval. The aim of channel coding is to allow transmission of information over a noisy channel in such a way that the receiver can recover the original information. To accomplish this, some level of redundancy is added to the information before transmission; this redundancy is added in such a way that it can be used to correct for the noise introduced by the channel. As a classic example of channel coding, we consider the case of compact discs. When storing music or data on a CD, one wants to ensure that the information can be retrieved even in the presence of scratches. Thus, it is necessary to include some redundancy with the information to guarantee that it is possible to recover the information in spite of the noise produced by these scratches. While the additional redundancy provides greater error-correction capabilities, it detracts from the amount of information that can be stored on the CD. Thus, a primary goal of coding theory is to design codes that minimize the amount of redundancy necessary to provide a certain level of error correction. Another focus of modern coding theory is to find decoding algorithms that will enable the user to efficiently recover the information from the erroneous data received. My research in algebraic coding theory thus far has consisted of three main projects. The first project has focused on understanding the particular class of efficient decoding algorithms known as iterative message-passing decoders. Specifically, the performance of these decoders is significantly impaired by the presence of pseudocodewords, and so we focused on investigating the structural properties of these error patterns. The second project has centered around the neuroscience model for how the brain “decodes” what stimulus it has encountered given a noisy neural response to that stimulus, a process known as neural coding. Therefore, we focused on understanding the potential links or parallels of that model to the coding theory model for communication in the presence of noise, with an eye toward using key coding theory principles to help explain some “paradoxical” neural coding phenomena. This project is farther afield from the typical coding theory topics, but involves an interesting application of the coding theory paradigm. Finally, I have also investigated the design of “good” codes for a new application of channel coding, namely the provision of error correction for random linear network coding. In particular, I have examined the connections between rank-metric and matrix codes as well as properties of these codes that determine their effectiveness for providing error correction in the network coding context. The first two research areas were the focus of two collaborative interdisciplinary research projects, and I will describe those briefly below. The last project has formed the bulk of my thesis work, and I will describe that in greater detail, together with some future work I have planned in that direction.