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Idempotents in Cyclic Codes

dc.contributor.advisorRobinson, Zacharyen_US
dc.contributor.authorBrame, Benjaminen_US
dc.contributor.departmentMathematicsen_US
dc.date.accessioned2012-05-20T15:21:01Z
dc.date.available2012-05-20T15:21:01Z
dc.date.issued2012en_US
dc.description.abstractCyclic codes give us the most probable method by which we may detect and correct data transmission errors. These codes depend on the development of advanced mathematical concepts. It is shown that cyclic codes, when viewed as vector subspaces of a vector space of some dimension n over some finite field F, can be approached as polynomials in a ring. This approach is made possible by the assumption that the set of codewords is invariant under cyclic shifts, which are linear transformations. Developing these codes seems to be equivalent to factoring the polynomial x[superscript]n-x over F. Each factor then gives us a cyclic code of some dimension k over F. Constructing factorizations of x[superscript]n-x is accomplished by using cyclotomic polynomials and idempotents of the code algebra. The use of these two concepts together allows us to find cyclic codes in F[superscript]n. Hence, the development of cyclic codes is a journey from codewords and codes to fields and rings and back to codes and codewords.en_US
dc.description.degreeM.A.en_US
dc.format.extent54 p.en_US
dc.format.mediumdissertations, academicen_US
dc.identifier.urihttp://hdl.handle.net/10342/3845
dc.language.isoen_US
dc.publisherEast Carolina Universityen_US
dc.subjectMathematicsen_US
dc.subjectCodesen_US
dc.subjectCodingen_US
dc.subjectCyclicen_US
dc.subjectCyclotomicen_US
dc.subject.lcshIdempotents
dc.subject.lcshCoding theory
dc.subject.lcshError-correcting codes (Information theory)
dc.titleIdempotents in Cyclic Codesen_US
dc.typeMaster's Thesisen_US

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