Y.S.Han Cyclic codes The polynomial g(x)is called the generator polynomial of the code. The degree of g(x)is equal to the number of parity-check digits of the code. The generator polynomial g(x)of an (n,k)cyclic code is a factor of x"+1. Proof:We have xg(x)=(n+1)+g(c). Since g(k)()is the code polynomial obtained by shifting g(x) to the right cyclically k times,g()is a multiple of g(). Hence, 2n+1={xk+a(x)}g(z). If g(x)is a polynomial of degree n-k and is a factor of x"+1, then g(x)generates an (n,k)cyclic code. School of Electrical Engineering Intelligentization,Dongguan University of TechnologyY. S. Han Cyclic codes 7 • The polynomial g(x) is called the generator polynomial of the code. • The degree of g(x) is equal to the number of parity-check digits of the code. • The generator polynomial g(x) of an (n, k) cyclic code is a factor of x n + 1. Proof: We have x k g(x) = (x n + 1) + g (k) (x). Since g (k) (x) is the code polynomial obtained by shifting g(x) to the right cyclically k times, g (k) (x) is a multiple of g(x). Hence, x n + 1 = {x k + a(x)}g(x). • If g(x) is a polynomial of degree n − k and is a factor of x n + 1, then g(x) generates an (n, k) cyclic code. School of Electrical Engineering & Intelligentization, Dongguan University of Technology