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