Weight distribution of some reducible cyclic codes

Abstract

Let q = p m where p is an odd prime, m ⩾ 3 , k ⩾ 1 and gcd ( k , m ) = 1 . Let Tr be the trace mapping from F q to F p and ζ p = e 2 π i p . In this paper we determine the value distribution of following two kinds of exponential sums ∑ x ∈ F q χ ( α x p k + 1 + β x 2 ) ( α , β ∈ F q ) and ∑ x ∈ F q χ ( α x p k + 1 + β x 2 + γ x ) ( α , β , γ ∈ F q ) , where χ ( x ) = ζ p Tr ( x ) is the canonical additive character of F q . As an application, we determine the weight distribution of the cyclic codes C 1 and C 2 over F p with parity-check polynomial h 2 ( x ) h 3 ( x ) and h 1 ( x ) h 2 ( x ) h 3 ( x ) , respectively, where h 1 ( x ) , h 2 ( x ) and h 3 ( x ) are the minimal polynomials of π −1 , π −2 and π − ( p k + 1 ) over F p , respectively, for a primitive element π of F q .