Weight distribution of some reducible cyclic codes

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 @z"p=e^2^@p^i^p. In this paper we determine the value distribution of following two kinds of exponential [email protected][email protected]?F"[email protected](@ax^p^^^k^+^[email protected]^2)(@a,@[email protected]?F"q) [email protected][email protected]?F"[email protected](@ax^p^^^k^+^[email protected]^[email protected])(@a,@b,@[email protected]?F"q), where @g(x)[email protected]"p^T^r^(^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 @p^-^1, @p^-^2 and @p^-^(^p^^^k^+^1^) over F"p, respectively, for a primitive element @p of F"q.