Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in [14,15,16]. In this paper we provide a slightly different approach toward the general problem and use it to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums to transform the problem of finding the weight distribution into a problem of evaluating certain character sums over finite fields, which on the special case is related with counting the number of points on some elliptic curves over finite fields. Other cases are also possible by this method.2000 Mathematics Subject Classification. 94B15,11T71,11T24.

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Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. Most previous results obtained so far were for cyclic codes with no more than three zeroes. Inspired by the works of Li et al. (Sci China Math 53:3279-3286, 2010; IEEE Trans Inf Theory 60:3903-3912, 2014), we study two families of cyclic codes over F p with arbitrary number of zeroes of generalized Niho type, more precisely C zeroes, and C(2) ( d 1 ,..., d t ) (for any prime p) of t zeroes for any t. We find that the first family has at most (2t + 1) non-zero weights, and the second has at most 2t non-zero weights. Their weight distribution are also determined in the paper.