Parity Check Polynomial

Let be a Cyclic Code with Generator Polynomial . The parity check polynomial is a polynomial such that

Corollary

Writing the Parity Check Matrix of is:

Proof

Just check that this is orthogonal with Generator Matrix given in Generator Polynomial.

Lemma

The Cyclic Code generated by , the reverse polynomial of , is Dual Code Alternatively, is the generator of the reverse code of