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.