On the Weight Distributions of Two Classes of Cyclic Codes
Jinquan Luo, Keqin Feng · IEEE Transactions on Information Theory · 2008
Let q=pmwhere p is an odd prime, mges2, and 1lesklesm-1. Let Tr be the trace mapping from Fqto Fpand zetap=e2pii/pbe a primitive pth root of unity. In this paper, we determine the value distribution of the following exponential sums: SigmaxisinFqchi(alphaxpk+1+betax2) (alpha, betaisinFq) where chi(x)=zetapTr(x)is the canonical additive character of Fq. As applications, we have the following. 1) We determine the weight distribution of the cyclic codes C1and C2over Fpt 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 pi-1, pi-2, and pi-(pk+1)over Fpt, respectively, for a primitive element pi of Fq. 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).