Several classes of cyclic codes with either optimal three weights or a few weights

Cyclic codes with a few weights are very useful in the design of frequency hopping sequences and the development of secret sharing schemes. In this paper, we mainly use Gauss sums to represent the Hamming weights of a general construction of cyclic codes. As applications, we obtain a class of optimal three-weight codes achieving the Griesmer bound, which generalizes a Vega’s result in \cite{V1}, and several classes of cyclic codes with only a few weights, which solve the open problem in \cite{V1}.