  In this paper we extend the works \cite{gegeng2,XLZD} further in two
directions and compute the weight distribution of these cyclic codes under more
relaxed conditions. It is interesting to note that many cyclic codes in the
family are optimal and have only a few non-zero weights. Besides using similar
ideas from \cite{gegeng2,XLZD}, we carry out some subtle manipulation of
certain exponential sums.
