Based on the weight properties of network codes, we present the refined versions of the Hamming bound, the Singleton bound and the Gilbert-Varshamov bound for linear network codes. We give two different algorithms to construct network codes with minimum distance constraints, both of which can achieve the refined Singleton bound. The first algorithm finds a codebook based on a given set of local encoding kernels defining a linear network code. The second algorithm finds a set of of local encoding kernels based on a given classical error-correcting code satisfying a certain minimum distance requirement. / First, the error correction/detection correction capabilities of a network code is completely characterized by a parameter which is equivalent to the minimum Hamming distance when the network code is linear and the weight measure on the error vectors is the Hamming weight. Our results imply that for a linear network code with the Hamming weight being the weight measure on the error vectors, the capability of the code is fully characterized by a single minimum distance. By contrast, for a nonlinear network code, two different minimum distances are needed for characterizing the capabilities of the code for error correction and for error detection. This leads to the surprising discovery that for a nonlinear network code, the number of correctable errors can be more than half of the number of detectable errors. (For classical algebraic codes, the number of correctable errors is always the largest integer not greater than half of the number of detectable errors.) / Network error correction provides a new method to correct errors in network communications by extending the strength of classical error-correcting codes from a point-to-point model to networks. This thesis considers a number of fundamental problems in coherent network error correction. / We further define equivalence classes of weight measures with respect to a general channel model. Specifically, for any given channel, the minimum weight decoders for two different weight measures are equivalent if the two weight measures belong to the same equivalence class. In the special case of network coding, we study four weight measures and show that all the four weight measures are in the same equivalent class for linear network codes. Hence they are all equivalent for error correction and detection when applying minimum weight decoding. / Yang, Shenghao. / Adviser: Raymond W.H. Yeung. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3708. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 89-93). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_344291 |
| Date | January 2008 |
| Contributors | Yang, Shenghao., Chinese University of Hong Kong Graduate School. Division of Information Engineering. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, theses |
| Format | electronic resource, microform, microfiche, 1 online resource (xi, 93 leaves : ill.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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