An important family of codes for error control in digital communications are the so-called cyclic codes; therefore, finding the weight distribution of a q-ary cyclic code C is not only a problem of theoretical interest, but also of practical importance. Typically, when the finite field q is a prime field, the problem is handled by expressing the Hamming weight of each codeword in C by means of certain combination of exponential sums. In this work, we will present a new method for computing the weight distribution of the dual of some cyclic codes with two non conjugated zeros. As we will see, such distribution is also given by means of the evaluation of certain exponential sums, however, such evaluation is only needed to be done over a subset. Moreover, this method has the advantage of flexibility, in the sense that it can also be applied to cyclic codes over finite fields of non prime order.
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