Combinatorial Methods in Coding Theory

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dc.contributor.advisorBarg, Alexanderen_US
dc.contributor.authorMazumdar, Aryaen_US
dc.contributor.departmentElectrical Engineeringen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.date.accessioned2011-07-06T05:57:03Z
dc.date.available2011-07-06T05:57:03Z
dc.date.issued2011en_US
dc.description.abstractThis thesis is devoted to a range of questions in applied mathematics and signal processing motivated by applications in error correction, compressed sensing, and writing on non-volatile memories. The underlying thread of our results is the use of diverse combinatorial methods originating in coding theory and computer science. The thesis addresses three groups of problems. The first of them is aimed at the construction and analysis of codes for error correction. Here we examine properties of codes that are constructed using random and structured graphs and hypergraphs, with the main purpose of devising new decoding algorithms as well as estimating the distribution of Hamming weights in the resulting codes. Some of the results obtained give the best known estimates of the number of correctable errors for codes whose decoding relies on local operations on the graph. In the second part we address the question of constructing sampling operators for the compressed sensing problem. This topic has been the subject of a large body of works in the literature. We propose general constructions of sampling matrices based on ideas from coding theory that act as near-isometric maps on almost all sparse signal. This matrices can be used for dimensionality reduction and compressed sensing. In the third part we study the problem of reliable storage of information in non-volatile memories such as flash drives. This problem gives rise to a writing scheme that relies on relative magnitudes of neighboring cells, known as rank modulation. We establish the exact asymptotic behavior of the size of codes for rank modulation and suggest a number of new general constructions of such codes based on properties of finite fields as well as combinatorial considerations.en_US
dc.identifier.urihttp://hdl.handle.net/1903/11547
dc.subject.pqcontrolledElectrical Engineeringen_US
dc.subject.pqcontrolledComputer Scienceen_US
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pquncontrolledCombinatoricsen_US
dc.subject.pquncontrolledCompressed Sensingen_US
dc.subject.pquncontrolledError-Correcting Codesen_US
dc.subject.pquncontrolledMemory and Storageen_US
dc.subject.pquncontrolledPermutation Codesen_US
dc.subject.pquncontrolledSparse Graph Codesen_US
dc.titleCombinatorial Methods in Coding Theoryen_US
dc.typeDissertationen_US

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