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)

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