We discuss the Carlitz-Uchiyama bound for the duals of BCH codes. An improvement of this bound was expected by MacWilliams and Sloane in their book The Theory of Error-Correcting Codes. We show that there exists a conjecturally infinite series of values such that for the designed distance of a BCH code belonging to that series this improvement is not true. On the contrary, we show that the Carlitz-Uchiyama bound is reached asymptotically. We deduce these results from corresponding results for exponential sums associated to monomials or to Dickson polynomials. We study also the extension of these results to the case of fields of characteristic different from 2.