On the Weight Distributions of Two Classes of Cyclic Codes
Let q = p(rn) 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 zeta(p) = e(2 pi i/p) be a primitive pth root of unity. In this paper, we determine the value distribution of the following exponential sums: Sigma(x epsilon Fq) chi (alpha x(pk+1) + beta x(2)) (alpha, beta epsilon F(q)) where chi(x) = zeta(Tr(x))(p) is the canonical additive character of F(q). As applications, we have the following. 1) We determine the weight distribution of the cyclic codes C(1) and C(2) over F(pt) with parity-check polynomial h(2)(x)h(3)(x) and h(1)(x) h(2)(x)h(3)(x), respectively, where t is a divisor of d = gcd (m, k), and h(1)(x), h(2)(x), and h(3)(x) are the minimal polynomials of pi(-1), pi(-2), and pi(-(pk+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).
weight distribution cyclic code exponential sum correlation distribution quadratic form