On the Number of Cyclic Codes Over \(\mathbb{Z}_{31}\)

Let n be a positive integer, yn - 1 cyclotomic polynomial and Zq be a given finite field. In this study we determined the number of cyclic codes over \(\mathbb{Z}_{31}\). First, we partitioned the cyclotomic polynomial yn - 1 using cyclotomic cosets 31 mod n and factorized yn - 1 using case to case basis. Each monic divisor obtained is a generator polynomial and generate cyclic codes. The results obtained are useful in the field of coding theory and more especially, in error correcting codes.