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Coding Theory via Groebner Bases
101 Seiten, Dissertation Technische Universität Hamburg-Harburg (2012), Softcover, A5
Coding theory plays an important role in efficient transmission of data over noisy communication channels. It consists of two steps; the first step is to encode the data to reduce its sensitivity to noise during transmission, and the second step is to decode the received data by detecting and correcting the noise induced errors. In this work an algebraic approach is used to develop efficient encoding and decoding algorithms for commonly used class of linear codes.
During this work the algebraic structure of linear codes is explored through the reduced Groebner basis. The advantage of this approach is that, once these Groebner bases are constructed standard procedure can be used to develop encoding and decoding processes. Furthermore, the binary and ternary Golay codes are studied algebraically. Finally, a presentation of the binomial ideal of a linear code in terms of its syzygy modules is provided and the corresponding finite free resolution has been described.