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Cyclic codes with a few weights are very useful in the design of frequencyhopping sequences and the development of secret sharing schemes. In this paper,we mainly use Gauss sums to represent the Hamming weights of a generalconstruction of cyclic codes. As applications, we obtain a class of optimalthree-weight codes achieving the Griesmer bound, which generalizes a Vega'sresult in \cite{V1}, and several classes of cyclic codes with only a fewweights, which solve the open problem in \cite{V1}.

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