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

Abstract

Let ℤ𝑞 be a finite field with 𝑞 element and 𝑥𝑛 −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 𝑥𝑛 −1 has been characterised. This paper will determine the number of irreducible monic polynomials and cyclotomic cosets of 𝑥𝑛 −1 over ℤ13 .The factorization of 𝑥𝑛 −1 over ℤ13 into irreducible polynomials using cyclotomic cosets of 13 modulo 𝑛 will be established. The number of irreducible polynomials factors of 𝑥𝑛 −1 over ℤ𝑞 is equal to the number of cyclotomic cosets of 𝑞 modulo 𝑛. Each monic divisor of 𝑥𝑛 −1 is a generator polynomial of cyclic code in 𝐹𝑞𝑛. 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 𝑥𝑛 −1. Finally, the number of cyclic codes of length 𝑛, when 𝑛= 13𝑘, 𝑛= 13𝑘, 𝑛= 13𝑘 −1, 𝑘, 13 = 1 are determine.