On the Weight Distributions of Two Classes of Cyclic Codes

Let q=p^m where p is an odd prime, mges2, and 1lesklesm-1. Let Tr be the trace mapping from F_q to F_p and zeta_p=e^2pii/p be a primitive pth root of unity. In this paper, we determine the value distribution of the following exponential sums: Sigma_{xisinF q}chi(alphax^{p k +1}+betax²) (alpha, betaisinF_q) where chi(x)=zeta_p ^Tr(x) is the canonical additive character of F_q. As applications, we have the following. 1) We determine the weight distribution of the cyclic codes C₁ and C₂ over F_pt with parity-check polynomial h₂(x)h₃(x) and h₁(x)h₂(x)h₃(x), respectively, where t is a divisor of d=gcd(m, k), and h₁(x), h₂(x) , and h₃(x) are the minimal polynomials of pi^-1, pi^-2, and pi^{-(p k +1)} over F_pt, respectively, for a primitive element pi of F_q. 2) We determine the correlation distribution between two m-sequences of period q-1. Moreover, we find a new class of p-ary bent functions. This paper extends the results in Feng and Luo (2008).