Complete weight enumerators of some cyclic codes

Let F_r be a finite field with r= q^m elements, α a primitive element of F_r, Tr _r/q the trace function from F_r onto F_q, r- 1 = nN for two integers n, N≥ 1 , and θ= α^N. In this paper, we use Gauss sums to investigate the complete weight enumerators of irreducible cyclic codes C = {c(a) = (Tr_r/q (a), Tr_r/q(aθ), . . . , Tr_r/q (aθ ⁿ⁻¹) : a ∈ F_r} and explicitly present the complete weight enumerators of some irreducible cyclic codes when gcd(n,q-1)=q-1 or q-1/2. Moreover, we determine the complete weight enumerators of a class of cyclic codes with the check polynomials h₁(x) h₂(x) by using Gauss sums, where h_i(x) are the minimal polynomials of α^-1_i over F_q and F^∗_q^mi=⟨αi⟩ for i= 1 , 2. We shall obtain their explicit complete weight enumerators if gcd (m₁, m₂) = 1 and q= 3 or 4.