On the weight distribution of a class of cyclic codes

Let q=pⁿ with n=2 m and p be a prime. Let 0lesklesn-1, knem. Assume k/(m, k) and m/(m, k) are both odd. In this paper we will study the following exponential sums of the Sigma_{xisinF q}zeta_p ^{Tr 1 m (alphax pm+1 )+Tr 1 n (betax pk+1 )} (alphaisinF_{p m}, betaisinF_q) where Tr₁ ⁿ:F_q-F_p and Tr₁ ^m:F_p ^mrarrF_p are the canonical trace mappings and zeta_p=e^2pii/p is a primitive p-th root of unity. As an application, we will determine the weight distribution of the cyclic codes C over F_p with parity-check polynomial h₁(x)h₂(x) where h₁(x) and h₂(x) are the minimal polynomials of pi^{-(p m +1)} and pi^{-(p k +1)} over F_p respectively for a primitive element pi of F_q.