A class of cyclic codes from two distinct finite fields

Let double-struck F_q be a finite field with q elements and m₁, m₂ two distinct positive integers such that gcd(m₁,m₂) = d. Suppose that α₁ and α₂ are two primitive elements of double-struck F_qm1 and double-struck F_qm2, respectively. Let n = (q^m1 - 1)(q^m2 - 1)/(q^d - 1) and T_i denote the trace function from double-struck F_qmi to double-struck F_q for i = 1,2. We define a cyclic code C_(q,m1,m2) = {c(a,b) : a ∈ double-struck F_qm1, b ∈ double-struck F_qm2}, where c(a,b) = (T₁(aα₁⁰) + T₂ (bα₂⁰), T₁(aα₁¹) + T₂(bα₂¹),..., T₁(aα₁^n-1) + T₂(bα₂^n-1)). In this paper, we use Gauss sums to investigate the weight distribution of C_(q,m1,m2) and prove that it has at most four nonzero weights if d = 2 and gcd(m₁-m₂/2, q - 1) = 1. Furthermore, we get a class of three-weight cyclic codes if |m₁-m₂| = 2. Some optimal or nearly optimal cyclic codes are presented.