Hamming Weights of the Duals of Cyclic Codes With Two Zeros

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In this paper, let Fr be a finite field with r = q^m. Suppose that g1, g2 ∈ F*_r are not conjugates over F_q, ord(g1) = n1, ord(g2) = n2, d = gcd(n1, n2), and n = n1n2/d. Let Fq(g1) = F_qm₁ , Fq(g2) = Fqm₂ , and Ti denote the trace function from Fq^m_i to Fq for i = 1, 2. We define a cyclic code C(q,m,n1,n2) = {c(a, b) : a ∈ F_qm₁ , b ∈ F_qm₂ }, where c(a, b) = (T₁(ag⁰₁) + T₂(bg⁰₂), T₁(ag¹₁) + T₂(bg¹₂), ... , T₁(ag^n-1₁ ) + T₂(bg^n-1₂ )). We mainly use Gauss periods to present the weight distribution of the cyclic code C(q,m,n1,n2). As applications, we determine the weight distribution of cyclic code C(q,m,qm_1-1,qm_2-1) with gcd(m1, m2) = 1; in particular, it is a three-weight cyclic code if gcd(q -1, m1 -m2) = 1. We also explicitly determine the weight distributions of some classes of cyclic codes including several classes of four-weight cyclic codes.