Idempotents in Cyclic Codes

ECU Author/Contributor (non-ECU co-authors, if there are any, appear on document)
Benjamin Brame (Creator)
Institution
East Carolina University (ECU )
Web Site: http://www.ecu.edu/lib/
Advisor
Zachary Robinson

Abstract: Cyclic 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.

Additional Information

Publication
Thesis
Date: 2012
Keywords
Mathematics, Code, Coding, Cyclic, Cyclotomic
Subjects
Idempotents
Coding theory
Error-correcting codes (Information theory)

Email this document to

This item references:

TitleLocation & LinkType of Relationship
Idempotents in Cyclic Codeshttp://hdl.handle.net/10342/3845The described resource references, cites, or otherwise points to the related resource.