Weight distribution of some reducible cyclic codes

Let q=pmq=pm where p   is an odd prime, m⩾3m⩾3, k⩾1k⩾1 and gcd(k,m)=1gcd(k,m)=1. Let Tr be the trace mapping from FqFq to FpFp and ζp=e2πip. In this paper we determine the value distribution of following two kinds of exponential sums∑x∈Fqχ(αxpk+1+βx2)(α,β∈Fq) and∑x∈Fqχ(αxpk+1+βx2+γx)(α,β,γ∈Fq), where χ(x)=ζpTr(x) is the canonical additive character of FqFq. As an application, we determine the weight distribution of the cyclic codes C1C1 and C2C2 over FpFp with parity-check polynomial h2(x)h3(x)h2(x)h3(x) and h1(x)h2(x)h3(x)h1(x)h2(x)h3(x), respectively, where h1(x)h1(x), h2(x)h2(x) and h3(x)h3(x) are the minimal polynomials of π−1π−1, π−2π−2 and π−(pk+1)π−(pk+1) over FpFp, respectively, for a primitive element π   of FqFq.