Idempotents in Cyclic Codes
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.
Brame, Benjamin. (January 2012). Idempotents in Cyclic Codes (Master's Thesis, East Carolina University). Retrieved from the Scholarship. (http://hdl.handle.net/10342/3845.)
Brame, Benjamin. Idempotents in Cyclic Codes. Master's Thesis. East Carolina University, January 2012. The Scholarship. http://hdl.handle.net/10342/3845. July 17, 2018.
Brame, Benjamin, “Idempotents in Cyclic Codes” (Master's Thesis., East Carolina University, January 2012).
Brame, Benjamin. Idempotents in Cyclic Codes [Master's Thesis]. Greenville, NC: East Carolina University; January 2012.
East Carolina University