On the Number of Cyclotomic Cosets and Cyclic Codes Over ℤ𝟏�𝟑�.

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 .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 modulo𝑛. Each monic divisor of 𝑥 𝑛 − 1 is a generator polynomial of cyclic code in 𝐹𝑞 𝑛. This paper will further demonstrate that the number of cyclic codes of length 𝑛 over a finite field 𝐹 is equal to the number of polynomials that divide𝑥 𝑛 − 1. Finally, the number of cyclic codes of length 𝑛, when 𝑛 = 13𝑘, 𝑛 = 13𝑘 , 𝑛 = 13𝑘 − 1, (𝑘, 13) = 1 are determined.