Algebraic Structure
Cyclic codes can be linked to ideals in certain rings. Let be a polynomial ring over the finite field . Identify the elements of the cyclic code C with polynomials in R such that maps to the polynomial : thus multiplication by x corresponds to a cyclic shift. Then C is an ideal in R, and hence principal, since R is a principal ideal ring. The ideal is generated by the unique monic element in C of minimum degree, the generator polynomial g. This must be a divisor of . It follows that every cyclic code is a polynomial code. If the generator polynomial g has degree d then the rank of the code C is .
The idempotent of C is a codeword e such that e2 = e (that is, e is an idempotent element of C) and e is an identity for the code, that is e c = c for every codeword c. If n and q are coprime such a word always exists and is unique; it is a generator of the code.
An irreducible code is a cyclic code in which the code, as an ideal, is maximal in R, so that its generator is an irreducible polynomial.
Read more about this topic: Cyclic Code
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