Weight distributions of cyclic codes with respect to pairwise coprime order elements

Let Fq be a finite field with q elements, where q = p , p is a prime, and s is a positive integer. An [n, k, d] linear code C is a k-dimensional subspace of Fq with minimum distance d. It is called cyclic if (c0, c1, . . . , cn−1) ∈ C implies (cn−1, c0, c1, . . . , cn−2) ∈ C. By identifying the vector (c0, c1, . . . , cn−1) ∈ Fq with c0 + c1x+ c2x 2 + · · ·+ cn−1c ∈ Fq[x]/(x − 1), any code C of length n over Fq corresponds to a subset of Fq[x]/(x − 1). Then C is a cyclic code if and only if the corresponding subset is an ideal of Fq[x]/(x n − 1). Note that every ideal of Fq[x]/(x n−1) is principal. Hence there is a monic polynomial g(x) with least degree such that C = 〈g(x)〉 and g(x) | (x − 1). Then g(x) is called the generator polynomial and h(x) = (x − 1)/g(x) is called the parity-check polynomial of the cyclic code C. Suppose that h(x) has u irreducible factors over Fq, we call C the dual of the cyclic code with u zeros. Let Ai be the number of codewords with Hamming weight i in the code C of length n. The weight enumerator of C is defined by