Cyclic Code

A code is a cyclic code if it is Linear Code and

We identify with the Ring via

Lemma

A code is cyclic if and only if satisfies

2. then

Corollary

is a cyclic code of length if and only if is an Ideal in We identify and .

Lemma

is a Principal Ideal Domain Moreover, each of its ideals is generated by some such that

Proof

Note that is Euclidean Domain so also Principal Ideal Domain. Now if is coprime with , then there is some such that So let be a generator of an ideal. Then is also in the ideal and divides the generator so is another generator. This way, we can get rid of factors of not dividing .

Corollary

We define the following Generator Polynomial Parity Check Polynomial

Proposition

If is odd, has no repeated roots so where are distinct irreducible polynomials in So number of cyclic codes of length is