Enumeration of cyclic codes over GF(17)

Abstract

In this paper we seek the number of irreducible polynomials of xn− 1 over GF(17). We factorize Xn− 1 over GF(17)into irreducible polynomials using cyclotomic cosets of 17 modulo n . The number of irreducible polynomials factors of Xn− 1 over fq is equal to the number of q cyclotomic cosets of modulo n. Each monic divisor of Xn− 1 is a generator polynomial of cyclic code in Fqn. We show that the number of cyclic codes of length n over a finite field f is equal to Xn− 1. Lastly, the number of cyclic codes of length n , when n= 17 , = the number of polynomials that divide 17k ,n = 17k,n=17k − 1, ( , 17) = 1 are enumerated.