Abstract:
Let Zq be a finite field with q element and x n − 1 be a given cyclotomic polynomial. The number of
cyclotomic cosets and cyclic codes has not been done in general. Although for different values of q the
polynomial x n − 1 has been characterised. This paper will determine the number of irreducible monic
polynomials and cyclotomic cosets of x n − 1 over Z13 .The factorization of x n − 1 over Z13 into irreducible
polynomials using cyclotomic cosets of 13 modulo n will be established. The number of irreducible
polynomials factors of x n − 1 over Zq is equal to the number of cyclotomic cosets of q modulo n. Each monic
divisor of x n − 1 is a generator polynomial of cyclic code in Fq n . This paper will further show that the
number of cyclic codes of length n over a finite field F is equal to the number of polynomials that divide
x n − 1. Finally, the number of cyclic codes of length n, when n = 13k, n = 13k , n = 13k − 1, k, 13 = 1
are determine.
