On the Weight Distributions of Two Classes of Cyclic Codes.

Let q = p^m where p is an odd prime, m ≥ 2, and 1 ≤ k ≤ m - 1. Let Tr be the trace mapping from F_q to F_p and (ζ_p = e 2π i/p be a primitive pth root of unity. In this paper, we determine the value distribution of the following exponential sums: Σ x ϵ F_q χ (αx^pk+1 + βx²) (α, β ϵ F_q) where _χ(x) = ζ_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_p^t 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 π^-1, π^-2, and π^-(Pk+1) over F_P^t respectively, for a primitive element π 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). [ABSTRACT FROM AUTHOR]