Weight distributions of two classes of cyclic codes with respect to two distinct order elements

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. Cyclic codes have been studied for many years, but their weight distributions are known only for a few cases. In this paper, let _r be an extension of a finite field _q and r = q ^m, we determine the weight distributions of the cyclic codes C = {c(a, b): a, b ∈ _r}, c(a, b) = (Tr_r/q(ag₁ + bg₂),.., Tr_r/q(ag₁^n-1 + bg ₂^n-1)), g₁, g₂ ∈ _r, in the following two cases: 1) ord(g₁) = n, n|r-1 and g₂ = 1 and 2) ord(g₁) = n, g₂ = g₁², ord(g₂) = n/2, m = 2, and 2(r-1)/n|(q + 1).